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Integers and opposites

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5225516
Write each banking amount as a signed number using \(+\) or \(-\). a) A credit of \(\$75\) b) A withdrawal of \(\$40\) c) An account balance of \(\$12\) d) A debt of \(\$100\)

Hints

- Decide whether each amount adds to or takes away from what is available. - Does the situation represent a value greater than or less than zero? - Which sign represents money you have, and which sign represents money you owe?

Solution

1. A credit adds money, so it is represented by \(+\$75\). 2. A withdrawal removes money, so it is represented by \(-\$40\). 3. Money available in the account is represented by \(+\$12\). 4. A debt is represented by \(-\$100\).

Answer

a) \(+\$75\) b) \(-\$40\) c) \(+\$12\) d) \(-\$100\)
5225596
Write each quantity as an integer with the appropriate sign. a) A desert basin is \(430\,\text{m}\) below sea level. b) A mountain summit is \(2962\,\text{m}\) above sea level. c) A research submersible is \(250\,\text{m}\) below the ocean surface. d) A freezer is \(18\) degrees Celsius below \(0\,^\circ\text{C}\).

Hints

- Identify the zero reference point in each situation. - Words such as “below” indicate one direction from zero. - Words such as “above” indicate the opposite direction.

Solution

1. Quantities below a reference level are negative, and quantities above a reference level are positive. 2. The basin is represented by \(-430\). 3. The mountain summit is represented by \(+2962\). 4. The submersible is represented by \(-250\). 5. The freezer temperature is represented by \(-18\).

Answer

a) \(-430\) b) \(+2962\) c) \(-250\) d) \(-18\)
5225816
An elevation of \(-45\,\text{m}\) means a location is \(45\,\text{m}\) below sea level. What does an elevation of \(+2962\,\text{m}\) mean? Also explain what \(0\,\text{m}\) represents in this context.

Hints

- What is the opposite of “below sea level”? - Picture a vertical number line through the water. - Where would the water surface be on that number line?

Solution

1. Negative elevations are below sea level, so positive elevations are above sea level. 2. An elevation of \(+2962\,\text{m}\) means the location is \(2962\,\text{m}\) above sea level. 3. An elevation of \(0\,\text{m}\) is the reference point: sea level.

Answer

An elevation of \(+2962\,\text{m}\) is \(2962\,\text{m}\) above sea level. The value \(0\,\text{m}\) represents sea level.
5411486
Insert either \(+\) or \(-\) in each blank so the two displayed integers are opposites. a) \(\square 54\) and \(-54\) b) \(+21\) and \(\square 21\)

Hints

- Opposite nonzero integers have the same distance from \(0\). - Their signs are different. - Keep the number part unchanged in each pair.

Solution

1. The opposite of \(-54\) is \(+54\), so part a) needs \(+\). 2. The opposite of \(+21\) is \(-21\), so part b) needs \(-\).

Answer

a) \(+\) b) \(-\)
5225676
Positive and negative numbers can represent opposite conditions or changes. Explain what each value means in its context. 1. A checking account balance is \(-\$150\). 2. A store's net income is \(-\$40\). 3. A diver's elevation is \(-25\,\text{m}\) relative to sea level. 4. The temperature changes by \(-8\,^\circ\text{C}\).

Hints

- Think about the opposite of a positive balance, a profit, an elevation above sea level, or an increase in temperature. - Identify what zero represents in each context. - Use the sign to determine the direction from zero.

Solution

1. A balance of \(-\$150\) means the account is overdrawn by \(\$150\). 2. Net income of \(-\$40\) means the store had a loss of \(\$40\). 3. An elevation of \(-25\,\text{m}\) means the diver is \(25\,\text{m}\) below sea level. 4. A change of \(-8\,^\circ\text{C}\) means the temperature decreased by \(8\,^\circ\text{C}\).

Answer

1. The account is overdrawn by \(\$150\). 2. The store had a loss of \(\$40\). 3. The diver is \(25\,\text{m}\) below sea level. 4. The temperature decreased by \(8\,^\circ\text{C}\).
5225776
The ideal temperature in a greenhouse is \(20\,^\circ\text{C}\). A gardener records these temperatures: Monday: \(23\,^\circ\text{C}\) Tuesday: \(19\,^\circ\text{C}\) Wednesday: \(20\,^\circ\text{C}\) Thursday: \(17\,^\circ\text{C}\) For each day, write the deviation from the ideal temperature as a signed number. Use positive numbers for temperatures above the ideal and negative numbers for temperatures below the ideal.

Hints

- Compare each measured temperature with the target temperature. - Which sign represents a value below the target? - What does a deviation of zero mean?

Solution

1. Monday is \(3\) degrees above the ideal, so the deviation is \(+3\,^\circ\text{C}\). 2. Tuesday is \(1\) degree below the ideal, so the deviation is \(-1\,^\circ\text{C}\). 3. Wednesday equals the ideal, so the deviation is \(0\,^\circ\text{C}\). 4. Thursday is \(3\) degrees below the ideal, so the deviation is \(-3\,^\circ\text{C}\).

Answer

Monday: \(+3\,^\circ\text{C}\) Tuesday: \(-1\,^\circ\text{C}\) Wednesday: \(0\,^\circ\text{C}\) Thursday: \(-3\,^\circ\text{C}\)
5225826
In a quiz game, correct answers add points and incorrect answers subtract points. A score of \(-10\) means a player is \(10\) points below zero. a) What does a score of \(+25\) mean? b) A player has a score of \(-5\). She answers a question correctly and earns \(5\) points. What is her new score, and what does it mean?

Hints

- If a negative score represents being below zero, what does a positive score represent? - What happens when a number and its opposite are combined? - Picture the scores on a number line.

Solution

1. A score of \(+25\) means the player is \(25\) points above zero. 2. Adding \(5\) to \(-5\) combines opposite numbers: \(-5+5=0\). 3. A score of \(0\) means the player is neither above nor below zero.

Answer

a) The player is \(25\) points above zero. b) The new score is \(0\), meaning the score is neither above nor below zero.
5411436
A theater sound board uses signed integers to show changes from a saved setting. Raising the level by seven steps is represented by \(+7\). a) Write the integer for lowering the level by seven steps. b) Explain why the two integers are opposites. c) What does \(0\) represent on this sound board?

Hints

- Identify which direction was chosen as positive. - Think about how an equal change in the reverse direction should be signed. - Ask what a change of zero would mean in this setting.

Solution

1. Lowering seven steps is the direction opposite raising seven steps, so it is represented by \(-7\). 2. The integers \(+7\) and \(-7\) have equal distance from \(0\) and opposite signs. 3. The integer \(0\) represents no change from the saved setting.

Answer

a) \(-7\) b) They are the same distance from \(0\) on opposite sides of \(0\). c) No change from the saved setting.
5411446
Devon says, “The opposite of the opposite of \(26\) is \(-26\).” Decide whether Devon is correct. Use signed integers to justify your decision.

Hints

- Carry out one “opposite” operation at a time. - Track the sign after the first operation before doing the second. - Compare the final number with the starting number.

Solution

1. The opposite of \(26\) is \(-26\). 2. The opposite of \(-26\) is \(26\), so \(-(-26)=26\). 3. Devon's final value has the wrong sign.

Answer

Devon is not correct. The opposite of the opposite of \(26\) is \(26\), because \(-(-26)=26\).
5411466
An integer \(p\) satisfies \(-p=-41\). a) Find \(p\). b) State the opposite of \(p\). c) Explain how the equation shows that your value is correct.

Hints

- Read the minus sign before the variable as “the opposite of.” - Work backward from the number on the right side. - Substitute your value into the original statement.

Solution

1. The equation says that the opposite of \(p\) is \(-41\). 2. Therefore, \(p=41\). 3. The opposite of \(41\) is \(-41\), matching the equation.

Answer

a) \(p=41\) b) \(-41\) c) Reversing the sign of \(41\) gives \(-41\), as required.
5411506
Jordan says, “Whenever I see a minus sign, the value must be negative.” Use the expression \(-(-8)\) to explain why this statement is not always true.

Hints

- Distinguish a negative number from an instruction to take an opposite. - Read the expression from the inside outward. - Determine the sign only after evaluating the opposite.

Solution

1. The outer minus sign means “take the opposite of \(-8\).” 2. The opposite of \(-8\) is \(8\). 3. Thus, \(-(-8)=8\), which is positive even though the expression contains minus signs.

Answer

Jordan's statement is false. In \(-(-8)\), the outer minus sign takes the opposite of \(-8\), so the value is \(8\), not a negative number.
5103786
Let \(x = -\frac{24}{6}\). a) Is \(x\) an integer? Is it rational? b) Where is the opposite of \(x\) located relative to zero on the number line? c) Give a rational number less than \(x\) that is not an integer.

Hints

- Simplify the fraction first. - Every integer can be written as a fraction with denominator \(1\). - Opposites are the same distance from zero on opposite sides. - Moving left on a number line gives smaller values.

Solution

1. Simplify: \(x = -\frac{24}{6} = -4\). 2. The number \(-4\) is an integer. Every integer is also rational because it can be written over \(1\). 3. The opposite of \(-4\) is \(4\), which lies to the right of zero. 4. A rational noninteger less than \(-4\) is \(-4.5\). Many other answers are possible.

Answer

a) \(x\) is an integer and a rational number. b) Its opposite is \(4\), to the right of zero. c) Answers will vary. One example is \(-4.5\).
5175546
Use the integers \(-12,7,0,-5\). a) Write the opposite of each integer in the given order. b) Combine the original integers and their opposites, remove duplicates, and order the resulting numbers from least to greatest using \(<\). c) List the pairs of different numbers that have the same absolute value.

Hints

- Taking an opposite changes the sign, except for zero. - Remove the repeated zero before ordering. - Different numbers with equal absolute value are opposites.

Solution

1. The opposites are \(12,-7,0,5\). 2. The distinct original numbers and opposites are \(-12,7,0,-5,12,-7,5\). In order, they are \(-12<-7<-5<0<5<7<12\). 3. The pairs of different numbers with equal absolute value are \((-12, 12)\), \((-7, 7)\), and \((-5, 5)\). Zero has no different opposite.

Answer

a) \(12,-7,0,5\) b) \(-12<-7<-5<0<5<7<12\) c) \((-12, 12)\), \((-7, 7)\), and \((-5, 5)\)
5217426
Use the numbers \(12,-18,5,-3,0,-25\). a) Write the opposite of each number in the given order. b) Order the original numbers from least to greatest using \(<\). c) Order the opposites from part a) from least to greatest.

Hints

- Opposite numbers are the same distance from zero on opposite sides. - On a number line, values increase from left to right. - Consider how taking opposites changes the order of a list.

Solution

1. Changing each number to its opposite gives \(-12,18,-5,3,0,25\). 2. Ordering the original numbers gives \(-25<-18<-3<0<5<12\). 3. Ordering the opposites gives \(-12<-5<0<3<18<25\).

Answer

a) \(-12,18,-5,3,0,25\) b) \(-25<-18<-3<0<5<12\) c) \(-12<-5<0<3<18<25\)
5226696
Start with the positive whole numbers \(1, 2, 3, \ldots\). a) Is the sum of any two positive whole numbers always a positive whole number? Is their product always a positive whole number? Give one example of each. b) Give an example showing that subtracting two positive whole numbers does not always produce a positive whole number. c) What numbers must be added to the positive whole numbers so that subtracting any two positive whole numbers always gives a number in the expanded set?

Hints

- Think about whether adding or multiplying counting numbers can take you outside the counting numbers. - Try subtracting a larger positive whole number from a smaller one. - Include the result of subtracting equal numbers and the numbers found to the left of \(0\) on a number line.

Solution

1. Addition and multiplication are closed for positive whole numbers. For example, \(4 + 7 = 11\) and \(3 \times 5 = 15\), and both results are positive whole numbers. 2. Subtraction is not closed for positive whole numbers. For example, \(3 - 8 = -5\), which is not positive. Also, subtracting a number from itself gives \(0\), which is not positive. 3. Add \(0\) and the negative integers \(-1, -2, -3, \ldots\). Together with the positive whole numbers, these form the integers.

Answer

a) Yes. For example, \(2 + 3 = 5\) and \(2 \times 3 = 6\). b) Answers will vary. One example is \(2 - 5 = -3\), which is not a positive whole number. c) Add \(0\) and all negative integers.
5411456
Mia claims, “Every integer has an opposite that is a different integer.” Is her claim always true? Give a counterexample and explain.

Hints

- Test the claim with several positive and negative integers. - Consider whether every integer is positive or negative. - Remember that one counterexample is enough to disprove an “every” statement.

Solution

1. Test the integer \(0\). 2. The opposite of \(0\) is \(0\), so the opposite is not a different integer. 3. Therefore, Mia's claim is false.

Answer

The claim is false. The integer \(0\) is a counterexample because its opposite is also \(0\).
5411476
Two signed integers describe equal changes in opposite directions. One integer is less than \(-30\) but greater than \(-35\), and its opposite is a multiple of \(8\). Find the two integers.

Hints

- List the few possible negative integers first. - Reverse each sign to see the corresponding opposite. - Test the positive values against the divisibility condition.

Solution

1. The integers between \(-35\) and \(-30\) are \(-34\), \(-33\), \(-32\), and \(-31\). 2. Their opposites are \(34\), \(33\), \(32\), and \(31\). 3. Only \(32\) is a multiple of \(8\), so the opposite pair is \(-32\) and \(32\).

Answer

\(-32\) and \(32\)
5411496
Use the set \(\{-31, 31, -9, 0, 12, 9\}\). a) List every pair of different integers in the set that are opposites. b) Which integer in the set is its own opposite? c) Which integer in the set has an opposite that is missing from the set?

Hints

- Group numbers by their distance from zero. - Check whether both signs appear for each nonzero distance. - Treat zero separately from the other entries.

Solution

1. The pairs with equal absolute value and opposite signs are \(\{-31, 31\}\) and \(\{-9, 9\}\). 2. Zero is its own opposite. 3. The opposite of \(12\) is \(-12\), which is missing.

Answer

a) \(\{-31, 31\}\) and \(\{-9, 9\}\) b) \(0\) c) \(12\)

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