Free math worksheets

- Absolute value is a number's distance from zero. - An absolute value cannot be negative. - Distinguish a negative sign outside the absolute-value bars from one inside them.

1. For a), both absolute values equal \(\frac{7}{8}\), so the statement is true. 2. For b), \(\left|\frac{5}{6}\right| = \frac{5}{6}\), so the negative sign outside gives \(-\frac{5}{6}\). The statement is false. 3. For c), \(\left|-\frac{1}{2}\right| = \frac{1}{2} > 0\), so the statement is true. 4. For d), \(\left|-\frac{3}{4}\right| = \frac{3}{4}\), not \(-\frac{3}{4}\), so the statement is false.

a) True b) False c) True d) False

- Absolute value represents distance from zero. - A positive distance from zero usually corresponds to two opposite integers. - Remember that zero is an integer.

1. The integers \(-17\) and \(17\) are both \(17\) units from zero. 2. The integers less than \(3\) units from zero are \(-2,-1,0,1,2\). 3. Only \(0\) is zero units from zero.

a) \(-17\) and \(17\) b) \(-2,-1,0,1,2\) c) \(0\)

- Think of points the same distance from zero on opposite sides. - Consider a number and its opposite. - Zero is the only integer equal to its own opposite.

1. Absolute value represents distance from zero. 2. A nonzero integer and its opposite are different points that are the same distance from zero. 3. For example, \(|5|=5\) and \(|-5|=5\), so the answer is yes.

Yes. A nonzero integer and its opposite have the same absolute value. For example, \(|12|=|-12|=12\).

- Find each number’s distance from zero. - Compare the two nonnegative distances. - The sign of the original number does not make its distance negative.

1. \(|-12|=12\) and \(|10|=10\). 2. Since \(12>10\), \(-12\) is farther from zero.

\(-12\) is farther from zero because \(|-12|=12>|10|=10\).

- Decide whether each pair lies on the same side of zero or on opposite sides. - Use distances from zero to determine each separation. - Compare the four distances.

1. In a), the points are on opposite sides of zero, so the distance is \(12+1=13\). 2. In b), both points are negative. Their distances from zero are \(13\) and \(1\), so the distance between them is \(13-1=12\). 3. In c), the points are on opposite sides of zero, so the distance is \(10+2=12\). 4. In d), the points are on opposite sides of zero, so the distance is \(5+7=12\). 5. The greatest distance is \(13\).

a) \(-12\) and \(1\)

- Two numbers can have the same positive distance from zero. - An absolute value cannot be negative. - Simplify the side without \(x\) first.

1. For a), absolute value is distance from zero. The two numbers at distance \(\frac{5}{12}\) from zero are \(\frac{5}{12}\) and \(-\frac{5}{12}\). 2. For b), absolute value is never negative, so there is no solution. 3. For c), first simplify the right side: \(\left\lvert-\frac{3}{7}\right\rvert = \frac{3}{7}\). Therefore, \(x = \frac{3}{7}\) or \(x = -\frac{3}{7}\).

a) \(x = \frac{5}{12}\) or \(x = -\frac{5}{12}\) b) No solution c) \(x = \frac{3}{7}\) or \(x = -\frac{3}{7}\)

- Absolute value represents distance from \(0\). - Two numbers can have the same nonzero distance from \(0\). - Add the two distances from zero to find the distance between opposite points.

1. Absolute value gives distance from \(0\), so the two numbers are \(5.2\) and \(-5.2\). 2. The points lie on opposite sides of zero, each \(5.2\) units away. Their distance is \(5.2+5.2=10.4\).

The numbers are \(-5.2\) and \(5.2\). The distance between them is \(10.4\).

- Think of distance from \(0\), not direction. - Include both positive and negative integers. - Do not forget \(0\).

1. Numbers less than \(4\) units from \(0\) have distances \(0\), \(1\), \(2\), or \(3\). 2. The integers are \(-3, -2, -1, 0, 1, 2, 3\). 3. There are \(7\) integers in the list.

The integers are \(-3, -2, -1, 0, 1, 2, 3\). There are \(7\) numbers.

- Compare the absolute values. - Convert fractions, decimals, and percents to a common form. - The smaller distance identifies the number closer to \(0\).

1. Distance from \(0\) is absolute value. 2. For a), \(\left|-\frac{5}{8}\right|=\frac{5}{8}=0.625\), while \(|0.65|=0.65\). Since \(0.625<0.65\), \(-\frac{5}{8}\) is closer to \(0\). 3. For b), \(|-1.2|=1.2\), and \(110\%=1.1\), so \(|110\%|=1.1\). Since \(1.1<1.2\), \(110\%\) is closer to \(0\).

a) \(-\frac{5}{8}\) b) \(110\%\)

- A midpoint is the same distance from both endpoints. - Find the distance from \(-5\) to each number in a pair. - Which pair gives equal distances?

1. For Pair A, the distance from \(-5\) to \(-12\) is \(7\), while the distance from \(-5\) to \(0\) is \(5\). The distances are not equal. 2. For Pair B, the distance from \(-5\) to \(-9\) is \(4\), and the distance from \(-5\) to \(-1\) is also \(4\). The distances are equal. 3. For Pair C, the distance from \(-5\) to \(-7\) is \(2\), while the distance from \(-5\) to \(-1\) is \(4\). The distances are not equal.

Pair B: \(-9\) and \(-1\)

- Use zero as an intermediate point to count the full distance. - The physical scale is \(1\,\text{cm}\) per unit interval. - The midpoint is half the total number of intervals from either endpoint.

1. From \(-14\) to \(0\) is \(14\) intervals, and from \(0\) to \(6\) is \(6\) more, for \(20\) intervals total. The segment is \(20\,\text{cm}\) long. 2. Half of \(20\) intervals is \(10\). Counting \(10\) intervals right from \(-14\) lands at \(-4\).

a) \(20\,\text{cm}\) b) \(-4\)

- Picture the temperatures on a vertical number line. - Use zero to split the total rise into two easy parts. - The halfway temperature is the midpoint of the two positions.

1. From \(-9\) to \(0\) is a rise of \(9\) degrees, and from \(0\) to \(3\) is \(3\) more, so the total rise is \(12\,\text{°F}\). 2. Half of \(12\) degrees is \(6\) degrees. Starting at \(-9\) and moving halfway toward \(3\) lands at \(-3\,\text{°F}\).

a) \(12\,\text{°F}\) b) \(-3\,\text{°F}\)

- Absolute value is distance from zero. - For part a), consider values on both sides of zero. - For part b), keep only negative integers.

1. Distances greater than \(4\) and less than \(8\) are \(5,6,7\). The corresponding integers are \(-7,-6,-5,5,6,7\). 2. The negative integers less than \(5\) units from zero are \(-4,-3,-2,-1\).

a) \(-7,-6,-5,5,6,7\) b) \(-4,-3,-2,-1\)

- Opposites have the same distance from zero but different signs. - Absolute value tells a number’s distance from zero. - Consider which number lies exactly on the center of the number line.

1. The number whose opposite is \(-25\) is \(25\). 2. The negative number with absolute value \(14\) is \(-14\). 3. The positive number with absolute value \(8\) is \(8\). 4. The only number satisfying \(x=-x\) is \(0\).

a) \(25\) b) \(-14\) c) \(8\) d) \(0\)

- Interpret absolute value as distance from zero. - Each nonzero distance corresponds to two opposite integers. - The endpoint values \(-6\) and \(6\) are not included.

1. The condition \(|z|<6\) means the number is fewer than \(6\) units from zero. 2. The negative integers are \(-5,-4,-3,-2,-1\), followed by \(0\) and the positive integers \(1,2,3,4,5\). 3. Therefore, the ordered list is \(-5,-4,-3,-2,-1,0,1,2,3,4,5\).

\(-5,-4,-3,-2,-1,0,1,2,3,4,5\)

- First identify the possible absolute values. - Each positive absolute value corresponds to a positive and a negative integer. - Keep only the negative integers.

1. The whole-number distances strictly between \(3\) and \(8\) are \(4,5,6,7\). 2. Because the integers must be negative, the solutions are \(-7,-6,-5,-4\).

\(-7,-6,-5,-4\)

- Absolute value represents distance from zero. - Use \(=\) for equal distances. - Use \(<\) when the first number is closer to zero.

1. Distance from zero is represented by absolute value. 2. Equal distances give \(|-14|=|14|\). 3. Being closer to zero means having a lesser absolute value, so \(|-5|<|-9|\). 4. Similarly, \(|2|<|-3|\).

a) \(|-14|=|14|\) b) \(|-5|<|-9|\) c) \(|2|<|-3|\)

- Find the absolute value of each number first. - Order the resulting distances from least to greatest. - Replace each distance with its original absolute-value expression.

1. The distances from zero are \(|-22|=22\), \(|17|=17\), and \(|-11|=11\). 2. Since \(11<17<22\), the required chain is \(|-11|<|17|<|-22|\).

\(|-11|<|17|<|-22|\)

- For part a), compare the numbers’ positions on a number line. - For part b), find each number’s distance from zero. - Two opposite numbers have equal absolute values.

1. Ordering the integers themselves gives \(-12<-8<-5<0<5<7\). 2. Their absolute values are \(12,7,5,0,5,8\), respectively. 3. Ordering by absolute value gives \(0,-5,5,7,-8,-12\). The positions of \(-5\) and \(5\) may be reversed because their absolute values are equal.

a) \(-12<-8<-5<0<5<7\) b) \(0,-5,5,7,-8,-12\); \(-5\) and \(5\) may appear in either order.

- The lesser number lies farther left. - Absolute value gives distance from zero. - Position and distance answer different questions.

1. In part a), \(-22<18\), so \(-22\) lies farther left. Also, \(|-22|=22>|18|=18\), so \(-22\) has the greater distance from zero. 2. In part b), \(-50<-45\), so \(-50\) lies farther left. Also, \(|-50|=50>|-45|=45\), so \(-50\) has the greater distance from zero. 3. In part c), \(-5<15\), so \(-5\) lies farther left. However, \(|15|=15>|-5|=5\), so \(15\) has the greater distance from zero.

a) Farther left: \(-22\); greater distance: \(-22\) b) Farther left: \(-50\); greater distance: \(-50\) c) Farther left: \(-5\); greater distance: \(15\)

- Evaluate each absolute value before comparing. - The open side of a comparison symbol faces the greater value. - Pay attention to whether a number is inside absolute-value bars.

1. Since \(-13\) lies to the right of \(-15\), \(-13>-15\). 2. Both \(|-18|\) and \(|18|\) equal \(18\), so they are equal. 3. Since \(|-7|=7\) and \(7>6\), \(|-7|>6\). 4. Since \(|-1|=1\) and \(0<1\), \(0<|-1|\).

a) \(>\) b) \(=\) c) \(>\) d) \(<\)

- Find each number’s distance from zero. - Put the lesser distance first. - Use absolute-value bars to name the distances.

1. The distances from zero are \(|-8|=8\) and \(|-6|=6\). 2. Since \(6<8\), \(-6\) is closer to zero, and \(|-6|<|-8|\).

\(-6\) is closer to zero, and \(|-6|<|-8|\).

- First list the possible absolute values. - Include the upper endpoint but not the lower endpoint. - For each positive distance, include both opposite integers.

1. The possible integer absolute values are \(8,9,10,11\). 2. Each positive absolute value corresponds to two opposite integers. 3. Therefore, the integers are \(-11,-10,-9,-8,8,9,10,11\).

\(-11,-10,-9,-8,8,9,10,11\)

- Absolute value is distance from zero. - Which whole-number distances lie strictly between \(99\) and \(103\)? - For each nonzero distance from zero, how many integer locations have that distance?

1. The only integer fewer than \(1\) unit from zero is \(0\). 2. The integer distances strictly between \(99\) and \(103\) are \(100\), \(101\), and \(102\). 3. Each positive distance corresponds to two integers, one on each side of zero. 4. Therefore, part b has the six solutions \(-102,-101,-100,100,101,102\).

a) \(0\) b) \(-102,-101,-100,100,101,102\)

- Absolute value is distance from zero. - Order the distances before returning to the original integers. - Write the original integers, not their absolute values, in part b).

1. The absolute values are \(34,12,5,0,42,17\). 2. Ordering those distances gives \(0<5<12<17<34<42\). 3. The corresponding original integers are \(0,-5,12,-17,-34,42\).

a) \(34,12,5,0,42,17\) b) \(0,-5,12,-17,-34,42\)

- List the possible whole-number distances first. - Each nonzero distance corresponds to a positive and a negative integer. - Neither endpoint is included.

1. The possible integer absolute values are \(13,14,15\). 2. Each positive absolute value corresponds to two opposite integers. 3. Therefore, the integers are \(-15,-14,-13,13,14,15\).

\(-15,-14,-13,13,14,15\)

- Decide whether each pair lies on the same side of zero or on opposite sides. - Across zero, add the two distances to zero. - On the same side, compare the magnitudes.

1. In a), both points are negative. Their distances from zero are \(7\) and \(22\), so the separation is \(22-7=15\). 2. In b), the segment crosses zero, so the distance is \(18+12=30\). 3. In c), the points are opposites, each \(55\) units from zero, so the distance is \(55+55=110\). 4. In d), both points are negative. Their magnitudes differ by \(140-95=45\), so the distance is \(45\).

a) \(15\) b) \(30\) c) \(110\) d) \(45\)

- Picture the numbers on a number line. - Absolute value measures distance from zero. - For part c, count units left from \(2\) and use the condition that the number is left of zero.

1. Zero is its own opposite. 2. The two numbers \(-12\) and \(12\) are both \(12\) units from zero. 3. For part c, start at \(2\) and move \(5\) units left on the number line. After \(2\) units you reach \(0\); moving \(3\) more units left reaches \(-3\). Its opposite is \(3\).

a) \(0\) b) \(-12\) and \(12\) c) The number is \(-3\), and its opposite is \(3\).

- Think of absolute value as a measured distance. - State what values an absolute value can have. - Compare that fact with the right side of the equation.

1. Absolute value represents distance from zero. 2. A distance is never negative, so \(|x|\ge0\) for every integer \(x\). 3. Therefore, \(|x|\) cannot equal \(-5\).

There is no integer solution because absolute value is always nonnegative, while \(-5\) is negative.

- Absolute value is distance from zero. - Find each absolute value before sorting. - Every absolute value is nonnegative.

1. The absolute values are \(24,13,35,7,2,0,19\). 2. From least to greatest, they are \(0<2<7<13<19<24<35\).

\(0,2,7,13,19,24,35\)

- Interpret \(|x|\) as the distance of \(x\) from zero. - First determine which whole-number distances lie strictly between \(2\) and \(5\). - For each positive distance, consider both points that are that far from zero.

1. The condition \(2<|x|<5\) means the distance of \(x\) from zero is greater than \(2\) units but less than \(5\) units. 2. The possible integer distances are \(3\) and \(4\). 3. Each positive distance gives two opposite integers, so \(x=-4,-3,3,4\).

\(x=-4,-3,3,4\); the possible distances from zero are \(3\) and \(4\) units.

- Reverse a number's direction from zero to find its opposite. - Compare the two points' distances from \(0\). - Think about the line of symmetry for an opposite pair on a number line.

1. The opposites are \(5.4\), \(-3.1\), \(0\), and \(\frac{1}{2}\), respectively. 2. A nonzero number and its opposite are the same distance from \(0\) on opposite sides. Zero is its own opposite. 3. Because each opposite pair is symmetric about \(0\), the point halfway between every pair is \(0\). The pair \(0\) and \(0\) also has midpoint \(0\).

1. \(5.4\), \(-3.1\), \(0\), and \(\frac{1}{2}\) 2. They are the same distance from \(0\) and lie on opposite sides, unless the number is \(0\). 3. The halfway point is \(0\) for every pair.

- Do not include the endpoints. - Compare each integer’s absolute value with \(|2|\). - Look for distances greater than \(2\).

1. The integers strictly between the endpoints are \(-5,-4,-3,-2,-1,0,1\). 2. The distance of \(2\) from zero is \(2\). 3. Among the listed integers, \(-5,-4,-3\) have absolute values greater than \(2\).

The integers are \(-5,-4,-3,-2,-1,0,1\). Of these, \(-5,-4,-3\) are farther from zero than \(2\).

- Test each statement with a positive number, a negative number, and \(0\). - Recall how opposite numbers are placed on a number line. - Interpret absolute value as distance from \(0\).

1. Statement a) is false. If \(x=-5\), then \(-x=5\), which is positive. If \(x=0\), then \(-x=0\). 2. Statement b) is true. If \(x\ne0\), the numbers \(x\) and \(-x\) lie on opposite sides of \(0\) at the same distance. If \(x=0\), they are the same point. In every case, \(|x|=|-x|\). 3. Statement c) is false. Absolute value is distance from \(0\), and \(|0|=0\).

a) False. For example, if \(x=-5\), then \(-x=5\). b) True. Opposite numbers have equal distance from \(0\). c) False. \(|0|=0\).

- Absolute value is never negative. - The number with the smaller absolute value is closer to \(0\). - Find the value halfway between the two distances. - A positive absolute value corresponds to two opposite numbers.

1. \(|a|=|-7|=7\) and \(|b|=|3|=3\). 2. Since \(3<7\), the number \(b=3\) is closer to \(0\). 3. The value halfway between \(3\) and \(7\) is \(\frac{3+7}{2}=5\). The numbers with absolute value \(5\) are \(-5\) and \(5\).

a) \(|a|=7\) and \(|b|=3\) b) \(b=3\) c) \(-5\) and \(5\)

- Locate each number between its two neighboring integers. - Decide which neighboring integer is closer. - Express the gap as a nonnegative distance.

1. For a), \(3.8\) is between \(3\) and \(4\) and is \(0.2\) unit from \(4\), so the distance is \(0.2\). 2. For b), \(-1\frac{1}{4}=-1.25\). It is one quarter unit from \(-1\), so the distance is \(0.25\). 3. For c), \(-\frac{22}{7}=-3\frac{1}{7}\). It is \(\frac{1}{7}\) unit to the left of \(-3\), so the distance is \(\frac{1}{7}\). 4. For d), \(0.45\) is closer to \(0\) than to \(1\), and its distance from \(0\) is \(0.45\).

a) \(0.2\) b) \(0.25\) c) \(\frac{1}{7}\) d) \(0.45\)

- First find how many integer intervals lie between the two values. - Use zero as an intermediate point when the values have different signs. - Each number-line interval represents \(2\,\text{cm}\).

1. In a), the segment crosses zero. There are \(5\) unit intervals from \(-5\) to \(0\) and \(8\) from \(0\) to \(8\), for \(13\) intervals total. The physical length is \(13\times2\,\text{cm}=26\,\text{cm}\). 2. In b), both points are left of zero. Their magnitudes differ by \(12-3=9\), so they are \(9\) unit intervals apart. The physical length is \(9\times2\,\text{cm}=18\,\text{cm}\).

a) \(26\,\text{cm}\) b) \(18\,\text{cm}\)

- A fixed distance can extend in either direction from a point. - Use zero as a reference when a move crosses from negative to positive. - Moving left from a negative number increases its distance from zero.

1. Moving \(25\) units to the right from \(-14\), it takes \(14\) units to reach \(0\) and \(11\) more units to reach \(11\). 2. Moving \(25\) units to the left makes the distance from zero \(14+25=39\), so the coordinate is \(-39\).

The two possible coordinates are \(11\) and \(-39\).

- Decide whether each segment crosses zero. - Across zero, combine the two distances to zero. - On the same side of zero, compare the two magnitudes.

1. From \(R=-18\) to \(S=4\), the segment crosses zero, so the distance is \(18+4=22\) units. 2. From \(S=4\) to \(T=22\), both points are on the positive side, so the distance is \(22-4=18\) units. 3. Since \(22>18\), point \(R\) is farther from \(S\). 4. From \(R=-18\) to \(T=22\), the segment crosses zero, so the distance is \(18+22=40\) units.

a) Point \(R\) is farther from \(S\): \(22\) units compared with \(18\) units. b) \(40\) units

- Picture each pair on a number line. - Find the total number of unit intervals between the endpoints. - The midpoint is half that many intervals from either endpoint.

1. In a), the endpoints are \(12\) unit intervals apart. Half is \(6\) intervals, and counting six intervals right from \(-14\) lands at \(-8\). 2. In b), the distance is \(3+9=12\) units across zero. Half is \(6\) units, and the halfway point is \(3\). 3. In c), the endpoints are opposites, so their midpoint is \(0\). 4. In d), the distance is \(10+20=30\) units across zero. Half is \(15\) units, placing the midpoint at \(5\).

a) \(-8\) b) \(3\) c) \(0\) d) \(5\)

- If \(M\) is the midpoint, the distances \(AM\) and \(MB\) are equal. - Use zero to find the distance from \(A\) to \(M\). - Continue the same distance and direction from \(M\).

1. From \(-6\) to \(0\) is \(6\) units, and from \(0\) to \(2\) is \(2\) more, so \(A\) and \(M\) are \(8\) units apart. 2. Because \(M\) is the midpoint, \(B\) must be \(8\) units on the other side of \(M\). 3. Counting \(8\) units right from \(2\) gives \(10\).

Point \(B\) is at \(10\).

- Translate each absolute-value condition into distances from zero. - “At most” includes the stated endpoint. - For part c), count zero once and each nonzero absolute value as an opposite pair.

1. The only whole-number absolute values strictly between \(9\) and \(12\) are \(10\) and \(11\). Therefore, \(z=-11,-10,10,11\). 2. The integers with \(|z|\le2\) are \(-2,-1,0,1,2\). 3. The integers with \(|z|<6\) are zero and the five opposite pairs with absolute values \(1\) through \(5\). Thus, there are \(1+5\times2=11\) integers.

a) \(-11,-10,10,11\) b) \(-2,-1,0,1,2\) c) \(11\)

- Organize each case by number, absolute value, and opposite. - Absolute value is distance from zero. - Opposites are the same distance from zero on different sides.

1. For \(-31\), the absolute value is \(31\), and the opposite is \(31\). 2. The negative number with absolute value \(19\) is \(-19\). Its opposite is \(19\). 3. If the opposite is \(50\), the original number is \(-50\). Its absolute value is \(50\). 4. Seven units to the right of zero is \(7\). Its absolute value is \(7\), and its opposite is \(-7\).

a) Number: \(-31\); absolute value: \(31\); opposite: \(31\) b) Number: \(-19\); absolute value: \(19\); opposite: \(19\) c) Number: \(-50\); absolute value: \(50\); opposite: \(50\) d) Number: \(7\); absolute value: \(7\); opposite: \(-7\)

- For part a), compare each number’s distance from zero with \(15\). - For part b), compare the original signed values, not their absolute values. - Among negative numbers, a number with greater absolute value is less.

1. The absolute values are \(22,14,5,30,35,9\). 2. The original numbers with absolute value greater than \(15\) are \(-22,30,-35\). 3. Ordering those numbers themselves from greatest to least gives \(30>-22>-35\).

a) \(-22,30,-35\) b) \(30>-22>-35\)

- Order the signed values separately from their absolute values. - Absolute value measures distance from zero. - Zero has the least possible absolute value.

1. Ordering the signed values gives \(-17<-14<-2<0<9<13\). 2. Their absolute values are \(14,9,2,0,13,17\). 3. The greatest absolute value is \(17\), which belongs to \(-17\). The least absolute value is \(0\), which belongs to \(0\).

a) \(-17<-14<-2<0<9<13\) b) \(-17\) c) \(0\)

- “Greatest to least” means descending order. - Check each number’s distance from zero for part b). - A value with absolute value exactly \(20\) does not qualify.

1. Ordering the signed values from greatest to least gives \(20>15>8>-2>-15>-38>-40\). 2. The absolute values are \(40,15,15,8,2,38,20\). 3. The numbers with absolute value strictly less than \(20\) are \(15,-15,8,-2\).

a) \(20>15>8>-2>-15>-38>-40\) b) \(15,-15,8,-2\)

- Opposite numbers have equal absolute values. - Write each absolute value before ordering. - Tied absolute values can appear in either order.

1. The absolute values are \(9,14,21,0,14,5,2\). 2. The numbers \(14\) and \(-14\) have the same absolute value. 3. Ordering the original numbers by absolute value gives \(0,-2,5,-9,14,-14,-21\). The positions of \(14\) and \(-14\) may be reversed.

a) \(14\) and \(-14\) b) \(0,-2,5,-9,14,-14,-21\); \(14\) and \(-14\) may appear in either order.

- First identify integers whose distance from zero exceeds \(4997\). - “Strictly between” excludes both endpoints. - Check the candidates against both conditions.

1. The integers with absolute value greater than \(4997\) near the interval endpoints include \(-5000,-4999,-4998\) and \(4998,4999,5000\). 2. Because the integers must lie strictly between \(-5000\) and \(5000\), both endpoints are excluded. 3. The remaining integers are \(-4999,-4998,4998,4999\).

\(-4999,-4998,4998,4999\)

- Test a positive integer, a negative integer, and zero. - Interpret absolute value as distance from zero. - Compare the cases \(x\ge0\) and \(x<0\).

1. If \(x\ge0\), its distance from zero equals its value, so \(|x|=x\). 2. If \(x<0\), then \(|x|\) is positive while \(x\) is negative, so \(|x|>x\). 3. Thus, \(|x|\) is either equal to or greater than \(x\), and it can never be less than \(x\).

a) All nonnegative integers, so \(x\ge0\) b) For \(x\ge0\), \(|x|=x\). For \(x<0\), \(|x|>x\). Therefore, \(|x|<x\) has no integer solutions.

- Find every number’s distance from zero. - The least distance identifies part a). - Keep absolute-value notation in the inequality chain.

1. The absolute values are \(12,5,8,2,1,10\). 2. The least absolute value is \(1\), which belongs to \(-1\). The greatest is \(12\), which belongs to \(-12\). 3. The ordered expressions are \(|-1|<|2|<|5|<|-8|<|10|<|-12|\).

a) \(-1\) b) \(-12\) c) \(|-1|<|2|<|5|<|-8|<|10|<|-12|\)

- Use zero as an intermediate point for distances between opposite signs. - Absolute value gives a number's distance from zero. - Compare the three absolute values in part c).

1. The points \(-32\) and \(12\) lie on opposite sides of zero, so their distance is \(32+12=44\). 2. The points \(12\) and \(-58\) lie on opposite sides of zero, so their distance is \(12+58=70\). 3. The distances from zero are \(|a|=32\), \(|b|=12\), and \(|c|=58\). Since \(58\) is greatest, \(c\) is farthest from zero.

a) \(44\) b) \(70\) c) \(c=-58\), because \(|-58|=58\) is the greatest absolute value.

- Interpret absolute value as distance from zero. - Think about the geometric positions of a number and its opposite. - Test an “always” statement with zero and a positive number as well as a negative number.

1. Statement a) is true. Absolute value is distance from zero, so the absolute value of an integer is a nonnegative integer. 2. Statement b) is true. Opposite nonzero numbers lie the same distance from zero in opposite directions, and zero is its own opposite. 3. Statement c) is false. For positive numbers and zero, the absolute value equals the number, such as \(|5|=5\).

a) True; an integer's absolute value is a nonnegative integer. b) True; opposite numbers have equal distance from zero and opposite directions, with \(0\) as its own opposite. c) False; for example, \(|5|=5\), not a greater value.

- A number’s distance from zero is its absolute value. - For ordering, picture the numbers from left to right on a number line. - For part c, compare the absolute values and choose the smallest distance.

1. The distances from zero are the absolute values. Since \(\lvert-14\rvert=14\) and \(\lvert11\rvert=11\), the numbers more than \(10\) units from zero are \(-14\) and \(11\). 2. From least to greatest, the numbers are \(-14<-6<-3<2<9<11\). 3. Compare the distances from zero: \(14,9,6,11,3,2\). The smallest is \(2\), so \(2\) is closest to zero.

a) \(-14\) and \(11\) b) \(-14<-6<-3<2<9<11\) c) \(2\)

- Split the total distance equally on both sides of \(0\). - Opposite numbers have \(0\) as their midpoint. - Apply the opposite operation one step at a time.

1. Opposite numbers are the same distance from \(0\). Each is \(15\div2=7.5\) units from \(0\). The ordered pairs are \((7.5, -7.5)\) and \((-7.5, 7.5)\). 2. The midpoint of opposite numbers is always \(0\), so \(r=0\). 3. The opposite of \(-8.3\) is \(8.3\), and the opposite of \(8.3\) is \(-8.3\).

1. \((7.5, -7.5)\) and \((-7.5, 7.5)\) 2. \(r=0\) 3. \(-8.3\)

- Picture equal distances extending in both directions from zero. - For \(|x|>12\), look beyond both \(-12\) and \(12\). - Use the definition of absolute value for the explanation.

1. The integers fewer than \(5\) units from zero are \(-4,-3,-2,-1,0,1,2,3,4\). 2. Being more than \(12\) units from zero means \(x\le-13\) or \(x\ge13\). 3. The statement is true because absolute value is defined as distance from zero.

1. \(-4,-3,-2,-1,0,1,2,3,4\) 2. \(x\le-13\) or \(x\ge13\) 3. True; absolute value measures distance from zero.

- Split the distance at \(0\) to find how far apart the endpoints are. - Move half of that distance from one endpoint to find the midpoint. - When counting integers between the endpoints, do not include the endpoints.

1. From \(-8\) to \(0\) is \(8\) units, and from \(0\) to \(4\) is \(4\) units, so the endpoints are \(12\) units apart. 2. Half of \(12\) is \(6\). Moving \(6\) units right from \(-8\) reaches \(-2\), so \(-2\) is the midpoint. 3. The integers strictly between the endpoints are \(-7,-6,-5,-4,-3,-2,-1,0,1,2,3\), for a total of \(11\).

The midpoint is \(-2\), and \(11\) integers lie strictly between the endpoints.

- Determine the value represented by one small tick interval. - Values left of zero are negative, and values right of zero are positive. - Count from a nearby labeled tick to each marker. - Absolute value is the distance from zero and is never negative.

1. There are \(5\) equal intervals between consecutive labeled multiples of \(10\), so each small interval represents \(2\). 2. Point \(X\) is at \(-24\), point \(Y\) is at \(-8\), and point \(Z\) is at \(16\). 3. Their distances from zero are \(|-24|=24\), \(|-8|=8\), and \(|16|=16\).

1. \(X=-24\), \(Y=-8\), and \(Z=16\) 2. \(|X|=24\), \(|Y|=8\), and \(|Z|=16\)

- Read the coordinate of each marked point. - For points on the same side of zero, compare their distances from zero. - For a segment that crosses zero, split its length at zero. - Compare the two segment lengths after finding them.

1. Read the coordinates: \(P=-12\), \(Q=-4\), and \(R=9\). 2. Points \(P\) and \(Q\) are on the same side of zero. Their distances from zero are \(12\) and \(4\), so \(PQ\) is \(12-4=8\,\text{cm}\). 3. Segment \(QR\) crosses zero. It is \(4\,\text{cm}\) from \(Q\) to zero and \(9\,\text{cm}\) from zero to \(R\), so \(QR\) is \(4+9=13\,\text{cm}\). 4. Since \(13-8=5\), segment \(QR\) is \(5\,\text{cm}\) longer.

Segment \(QR\) is \(5\,\text{cm}\) longer than segment \(PQ\).

- Values left of zero are negative. - A number's distance from zero is its absolute value.

1. Each small tick interval represents \(1\). The marked coordinates are \(A=-17\), \(B=-8\), and \(C=4\). 2. Their distances from zero are \(|-17|=17\), \(|-8|=8\), and \(|4|=4\). 3. Since \(17\) is greatest, \(A=-17\) is farthest from zero.

a) \(A=-17\), \(B=-8\), \(C=4\) b) \(A=-17\) is farthest from zero.

- Convert the physical length to number-line units first. - Each endpoint is the same distance from the midpoint. - Count the same number of units left and right from \(-5\).

1. A \(6\,\text{cm}\) segment represents \(6\times2=12\) number-line units. 2. Each endpoint is half of \(12\), or \(6\) units, from the midpoint. 3. Counting \(6\) units left from \(-5\) gives \(-11\). Counting \(6\) units right from \(-5\) gives \(1\).

The endpoints are \(-11\) and \(1\).

- A nonzero distance from zero corresponds to two opposite numbers. - Think about what taking an opposite does to a sign. - A distance cannot be negative.

1. The two numbers \(-21\) and \(21\) are both \(21\) units from zero. 2. If the opposite of \(x\) is \(-4\), then \(x=4\), and \(|x|=4\). 3. The statement is true because absolute value represents distance, and distance cannot be negative. 4. Since \(|-3|=3\), the number must have absolute value \(0\), \(1\), or \(2\). It cannot be \(-1\) or \(-2\), so \(0\), \(1\), or \(2\) works.

a) \(-21\) and \(21\) b) \(x=4\) and \(|x|=4\) c) True; absolute value is a distance and is therefore nonnegative. d) One possible answer is \(0\). Other possible answers are \(1\) and \(2\).

- A counterexample satisfies the condition but makes the conclusion false. - Consider choosing a negative value for \(a\). - A number can have a large absolute value and still be less than a positive number.

1. Choose \(a=-10\) and \(b=5\). 2. Then \(|a|=10\) and \(|b|=5\), so \(|a|=2|b|\). 3. However, \(-10<5\), so \(a<b\). This counterexample disproves the claim.

One counterexample is \(a=-10\) and \(b=5\). Then \(|a|=10=2\times5=2|b|\), but \(a<b\) because \(-10<5\).

- Find the opposite and absolute value of each example separately. - Compare what happens to positive and negative signs. - Remember that \(0\) is its own opposite.

1. For \(+7\), removing the plus sign leaves \(7\), but the opposite is \(-7\). Leon’s rule fails. 2. For \(-12\), removing the negative sign gives \(12\), which happens to be both the opposite and the absolute value of \(-12\). 3. The opposite of \(0\) is \(0\), and \(|0|=0\). 4. To find an opposite, change the sign, except that \(0\) remains \(0\). Absolute value is distance from zero and is never negative.

Sophie is correct. The opposite is found by changing a number’s sign, with \(0\) as its own opposite. Absolute value is the number’s distance from zero. Removing a negative sign gives the absolute value only when the number is written as negative.

- For the counterexample, try one negative and one positive integer. - Among negative integers, the lesser value lies farther left. - Equal absolute values mean equal distances from zero.

1. For example, let \(a=-10\) and \(b=2\). Then \(-10<2\), but \(|-10|=10>|2|=2\), so the claim is false. 2. Because \(a<b<0\), \(a\) lies farther left and therefore farther from zero. Thus, \(|a|>|b|\). 3. Two different numbers with equal absolute value lie on opposite sides of zero at equal distances. They are opposites.

a) One counterexample is \(a=-10\), \(b=2\): \(a<b\), but \(|a|>|b|\). b) \(a\) has the greater absolute value because it lies farther from zero. c) The points are symmetric about zero.