Aimathic – One-step inequalities and solution sets

One-step inequalities and solution sets

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Difficulty: 1 – 5

5496756

Write an inequality for this condition: A puzzle takes more than $12$ minutes to solve. Let $m$ be the number of minutes.

Hints

- Identify the boundary value in the condition. - Decide whether allowed values lie above or below that boundary.

Solution

1. “More than” indicates values greater than the boundary. 2. The boundary is $12$. 3. The inequality is $m>12$.

Answer

$m>12$

5496766

Write an inequality for this condition: A storage bin holds fewer than $20$ notebooks. Let $n$ be the number of notebooks.

Hints

- Connect the comparison words to the position of allowed values. - Check whether the boundary amount itself is included by the wording.

Solution

1. “Fewer than” indicates values less than the boundary. 2. The boundary is $20$. 3. The inequality is $n<20$.

Answer

$n<20$

5496826

From the set of whole numbers $\{0,1,2,3,4,5,6\}$, list all solutions to $w<4$.

Hints

- Use the specified set rather than listing every possible number. - Stop before the boundary because the comparison is strict.

Solution

1. Select whole numbers less than $4$. 2. The solutions are $0$, $1$, $2$, and $3$. 3. The value $4$ is not included.

Answer

$\{0,1,2,3\}$

5496946

Describe how to represent all solutions to $x<2$ on a number line. Include what happens at $2$ and which direction the solution set extends.

Hints

- Decide whether the boundary value belongs to a strict inequality. - Recall which direction contains numbers less than the boundary.

Solution

1. The inequality is strict, so $2$ is not a solution. 2. Represent the excluded boundary with an open circle at $2$. 3. Values less than $2$ lie to the left, so the solution set is a ray extending left from $2$.

Answer

Place an open circle at $2$ and shade a ray extending to the left.

5496996

Describe the solution set of $p>0.4$ in words. Include what happens at $p=0.4$.

Hints

- Translate the symbol into a complete sentence. - State explicitly whether the boundary belongs to the set.

Solution

1. The solutions are all numbers greater than $0.4$. 2. The value $0.4$ itself makes $0.4>0.4$, which is false. 3. Therefore, the boundary is not included.

Answer

All numbers greater than $0.4$, not including $0.4$.

5169896

Find every digit $d$ from $0$ through $9$ that makes each inequality true. a) $40d < 250$ b) $700d > 4000$ c) $90d < 800$

Hints

- Use nearby multiplication facts to locate the boundary value. - Check the digits immediately below and above that boundary. - Pay close attention to whether each inequality uses $<$ or $>$.

Solution

1. For a), $40 \times 6 = 240$ and $40 \times 7 = 280$. Therefore, the solutions are the digits $0$ through $6$. 2. For b), $700 \times 5 = 3500$ and $700 \times 6 = 4200$. Therefore, the solutions are $6, 7, 8$, and $9$. 3. For c), $90 \times 8 = 720$ and $90 \times 9 = 810$. Therefore, the solutions are the digits $0$ through $8$.

Answer

a) $0, 1, 2, 3, 4, 5, 6$ b) $6, 7, 8, 9$ c) $0, 1, 2, 3, 4, 5, 6, 7, 8$

5200226

Which multiples of $10$ from $10$ through $90$ make this inequality true? $6x > 400$ Check your choices by relating each product to a basic multiplication fact.

Hints

- Compare each product with $400$ using related basic multiplication facts. - Estimate the boundary value by dividing $400$ by $6$. - Once one multiple of $10$ works, decide whether larger multiples also work. - More than one value may work.

Solution

1. Test the nearby multiple of $10$: $6 \times 60 = 360$, which is not greater than $400$. 2. The next value works because $6 \times 70 = 420 > 400$. 3. Larger choices also work: $6 \times 80 = 480$ and $6 \times 90 = 540$. 4. Therefore, the values are $70$, $80$, and $90$.

Answer

$x = 70$, $80$, or $90$.

5227136

Find all integers $z$ that satisfy $-7<z\le -2$.

Hints

- Identify the integers to the right of $-7$. - Pay attention to which endpoint is included. - Test each integer against both parts of the inequality.

Solution

1. The condition $z>-7$ excludes $-7$, so the least possible integer is $-6$. 2. The condition $z\le -2$ includes $-2$. 3. Therefore, the integers are $-6,-5,-4,-3,-2$.

Answer

$-6,-5,-4,-3,-2$

5227146

Which numbers in the list $-12,-8,-5,-2,0,3$ satisfy $-9<n<-1$?

Hints

- Picture the interval between $-9$ and $-1$ on a number line. - Test each listed number against both inequalities. - The endpoints are not included.

Solution

1. A qualifying number must be greater than $-9$ and less than $-1$. 2. Testing the listed values shows that $-8$, $-5$, and $-2$ meet both conditions.

Answer

$-8,-5,-2$

5496776

A climbing wall route is open only to students taller than $48$ inches. Let $h$ be a student’s height in inches. Write the inequality and state whether $h=48$ is a solution.

Hints

- Translate the comparison before testing the boundary. - A strict comparison does not include equality.

Solution

1. “Taller than $48$” gives $h>48$. 2. Substitute $48$: the statement $48>48$ is false. 3. Therefore, $h=48$ is not a solution.

Answer

Inequality: $h>48$ $h=48$ is not a solution.

5496786

A freezer alarm turns on when the temperature $t$ is below $-5$ degrees Celsius. Write an inequality for alarm temperatures.

Hints

- Place the boundary on a number line mentally. - Temperatures below a negative number lie farther to the left.

Solution

1. “Below” means less than on the number line. 2. The boundary is $-5$. 3. Alarm temperatures satisfy $t<-5$.

Answer

$t<-5$

5496796

A team’s supply cost $c$ must stay under $\$25$. Which inequality represents the condition: $c>25$, $c<25$, or $25<c$?

Hints

- Translate “under” into a comparison with the boundary. - Read each proposed inequality from left to right before choosing.

Solution

1. “Under $25$” means less than $25$. 2. The matching inequality is $c<25$. 3. The expression $25<c$ is equivalent to $c>25$, so it does not match.

Answer

$c<25$

5496806

Which values in the set $\{2.9,3.5,3.6,4,5.2\}$ satisfy $x>3.5$?

Hints

- Test the listed values one at a time against the same boundary. - Do not include a value equal to the boundary in a strict inequality.

Solution

1. Compare each value with $3.5$. 2. The values strictly greater than $3.5$ are $3.6$, $4$, and $5.2$. 3. The boundary value $3.5$ is excluded.

Answer

$\{3.6,4,5.2\}$

5496836

The labeled points are possible values of $n$. Which labels mark solutions to $n>4$?

Hints

- Locate the boundary first, then compare each labeled point by position. - Points equal to the boundary do not satisfy a strict inequality.

Solution

1. The labeled values are $2$, $4$, $5$, and $7$. 2. Values to the right of $4$ are greater than $4$. 3. Labels C and D mark $5$ and $7$, so they are the solutions.

Answer

C and D

5496846

The labeled points are possible temperatures. Which labels satisfy $t<-1.5$?

Hints

- Use left-to-right order on the number line for negative values. - The marker at the boundary is not included in a strict solution set.

Solution

1. The marked values are $-3$, $-1.5$, $-1$, and $0.5$. 2. Only values to the left of $-1.5$ are less than $-1.5$. 3. Label A at $-3$ is the only solution.

Answer

5496866

Riley says $6$ is a solution to $x>6$ because it is the boundary shown in the inequality. Is Riley correct? Explain.

Hints

- Test the claimed value by replacing the variable. - Pay attention to whether the comparison symbol includes equality.

Solution

1. Substitute $x=6$: the statement becomes $6>6$. 2. A number is not greater than itself, so the statement is false. 3. Riley is not correct; $6$ is not a solution.

Answer

No. $6>6$ is false, so the boundary is not a solution.

5496886

A fish must be shorter than $7.5$ centimeters to fit through an opening. Let $l$ be its length. Write the inequality and give three possible decimal lengths.

Hints

- Write the comparison before choosing examples. - Check every example against the same strict boundary.

Solution

1. “Shorter than $7.5$” gives $l<7.5$. 2. Any nonnegative decimal below $7.5$ is a possible length. 3. Examples are $7.4$, $6.8$, and $3.25$ centimeters.

Answer

$l<7.5$. Example solutions: $7.4\,\text{cm}$, $6.8\,\text{cm}$, and $3.25\,\text{cm}$.

5496896

What is the least whole-number solution to $n>7.2$? Explain why the previous whole number does not work.

Hints

- Locate the decimal boundary between two consecutive whole numbers. - Choose the first whole number on the allowed side of the boundary.

Solution

1. Whole numbers greater than $7.2$ begin with $8$. 2. The previous whole number is $7$, and $7>7.2$ is false. 3. Therefore, the least whole-number solution is $8$.

Answer

$8$. The previous whole number, $7$, does not work because $7>7.2$ is false.

5496906

What is the greatest integer solution to $z<-2.4$?

Hints

- Place the boundary between neighboring integers on a number line. - For negative integers, the value farther right is greater.

Solution

1. Integers less than $-2.4$ include $-3,-4,-5,\ldots$. 2. Among these, $-3$ is farthest to the right. 3. Therefore, the greatest integer solution is $-3$.

Answer

$-3$

5496916

The table lists possible values. Mark each statement true or false for the inequality $q<1.25$. <table> <thead> <tr> <th>Value of $q$</th> <th>Satisfies?</th> </tr> </thead> <tbody> <tr> <td>$1.2$</td> <td>?</td> </tr> <tr> <td>$1.25$</td> <td>?</td> </tr> <tr> <td>$1.3$</td> <td>?</td> </tr> <tr> <td>$0.9$</td> <td>?</td> </tr> </tbody> </table>

Hints

- Compare decimal place values carefully around the boundary. - Test equality separately because the symbol is strict.

Solution

1. $1.2<1.25$ is true. 2. $1.25<1.25$ is false. 3. $1.3<1.25$ is false. 4. $0.9<1.25$ is true.

Answer

$1.2$: true $1.25$: false $1.3$: false $0.9$: true

5496926

A package must weigh under $70$ pounds. Let $w$ be its weight. Write the inequality. Then decide whether weights $69.9$, $70$, and $70.1$ pounds are allowed.

Hints

- Translate the word “under” before checking the test values. - Compare values just below, equal to, and just above the boundary.

Solution

1. The condition is $w<70$. 2. $69.9<70$ is true, so $69.9$ pounds is allowed. 3. $70<70$ and $70.1<70$ are false, so those weights are not allowed.

Answer

Inequality: $w<70$ Allowed: $69.9\,\text{lb}$ Not allowed: $70\,\text{lb}$, $70.1\,\text{lb}$

5496956

Use substitution to determine which of $-4$, $-1$, and $2$ satisfy $r>-2$.

Hints

- Replace the variable with each candidate and read the resulting comparison. - Use number-line order for the negative candidates.

Solution

1. $-4>-2$ is false. 2. $-1>-2$ is true. 3. $2>-2$ is true. 4. The solutions from the list are $-1$ and $2$.

Answer

$-1$ and $2$

5496966

The labeled points are possible values of $x$. Which labels satisfy $x<\frac{3}{4}$?

Hints

- Use the spatial order of the marked fractions. - Exclude the point located exactly at the boundary.

Solution

1. The marked values are $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, and $1$. 2. Values left of $\frac{3}{4}$ are smaller. 3. Labels A and B are solutions.

Answer

A and B

5496976

The labeled decimals are possible race times in minutes. Which labels satisfy $t>2.6$?

Hints

- Locate the decimal boundary on the scale. - Use rightward position for “greater than,” excluding the boundary marker.

Solution

1. The values are $2.4$, $2.6$, $2.7$, and $3.0$. 2. Values strictly to the right of $2.6$ satisfy the inequality. 3. Labels Y and Z mark $2.7$ and $3.0$, so they are the solutions.

Answer

Y and Z

5496986

A youth orchestra accepts musicians younger than $13$. Let $a$ be age in years. Write the inequality and explain whether a musician who is exactly $13$ meets the condition.

Hints

- Translate the age comparison directly. - Check the exact boundary rather than assuming it is included.

Solution

1. “Younger than $13$” gives $a<13$. 2. At $a=13$, the comparison $13<13$ is false. 3. A musician exactly $13$ does not meet the condition.

Answer

$a<13$. A musician exactly $13$ does not meet the condition.

5103756

Find all whole-number values of $k$ that make the inequality true: $\frac{k}{12}<\frac{3}{4}$

Hints

- Rewrite the boundary fraction with denominator $12$. - With equal positive denominators, compare the numerators. - List every whole number below the boundary.

Solution

1. Rewrite the boundary with denominator $12$: $\frac{3}{4}=\frac{9}{12}$. 2. The inequality becomes $\frac{k}{12}<\frac{9}{12}$, so $k<9$. 3. The whole-number solutions are $0,1,2,3,4,5,6,7,8$.

Answer

$k\in\{0,1,2,3,4,5,6,7,8\}$

5169906

Let $d$ be a digit in $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. Which digits satisfy both inequalities? $30d > 100$ $80d < 500$

Hints

- Solve each inequality separately. - List the digit solutions for the first inequality. - List the digit solutions for the second inequality. - Keep only the digits that appear in both lists.

Solution

1. For $30d > 100$, $30 \times 3 = 90$ and $30 \times 4 = 120$, so the digit solutions are $4, 5, 6, 7, 8$, and $9$. 2. For $80d < 500$, $80 \times 6 = 480$ and $80 \times 7 = 560$, so the digit solutions are $0, 1, 2, 3, 4, 5$, and $6$. 3. The digits in both solution sets are $4, 5$, and $6$.

Answer

$4, 5, 6$

5174856

Find all whole numbers $x$ that satisfy both conditions. 1. The whole number immediately after $x$ is greater than $3$. 2. The whole number immediately before $x$ is less than $6$.

Hints

- Translate the number immediately after $x$ as $x+1$. - Translate the number immediately before $x$ as $x-1$. - List the whole numbers common to both solution sets.

Solution

1. The whole number after $x$ is $x+1$. The condition $x+1>3$ means $x>2$. 2. The whole number before $x$ is $x-1$. The condition $x-1<6$ means $x<7$. 3. The whole numbers satisfying both conditions are $3,4,5,6$.

Answer

$x\in\{3,4,5,6\}$

5496816

Among fractions with denominator $12$, find the greatest value that satisfies $y<\frac{2}{3}$. Explain why the next greater fraction with denominator $12$ is not a solution.

Hints

- Express the boundary using the required denominator. - Pay attention to whether equality at the boundary is allowed.

Solution

1. Rewrite the boundary as $\frac{2}{3}=\frac{8}{12}$. 2. Because the inequality is strict, $\frac{8}{12}$ is not included. 3. The greatest fraction with denominator $12$ below $\frac{8}{12}$ is $\frac{7}{12}$.

Answer

$\frac{7}{12}$. The next greater fraction with denominator $12$ is $\frac{8}{12}=\frac{2}{3}$, which is excluded by $<$.

5496856

Explain why $x>10$ has infinitely many solutions. Give three different solutions, including one non-whole number.

Hints

- Think beyond whole numbers when considering possible solutions. - Ask whether there is any final or greatest number allowed by the condition.

Solution

1. Every number greater than $10$ satisfies the inequality. 2. There is no largest number, and numbers can be chosen arbitrarily close to or far above $10$. 3. Examples include $11$, $10.5$, and $100$.

Answer

There are infinitely many numbers greater than $10$. For example, $10.5$, $11$, and $100$ are solutions.

5496876

A sign says, “Participants must be older than $11$.” A student writes $a\ge 11$. Explain why this does not match the wording and write the correct inequality.

Hints

- Compare the meaning of the words with whether equality is allowed. - Test the boundary age against the condition.

Solution

1. “Older than $11$” excludes age $11$. 2. The symbol $\ge$ would include $11$, so it does not match. 3. The correct inequality is $a>11$.

Answer

The inequality $a\ge 11$ incorrectly includes age $11$. The correct inequality is $a>11$.

5496936

Do the inequalities $x>5$ and $x<5$ share any solutions? Does $x=5$ satisfy either one?

Hints

- Compare the directions of the two conditions from the same boundary. - Test the boundary separately in both statements.

Solution

1. A number cannot be both greater than $5$ and less than $5$. 2. Therefore, the two solution sets do not overlap. 3. At $x=5$, both $5>5$ and $5<5$ are false.

Answer

They share no solutions, and $x=5$ satisfies neither inequality.

5497006

Give one fraction between $\frac{5}{8}$ and $1$ that satisfies $x>\frac{5}{8}$. Verify it.

Hints

- Choose a fraction on the allowed side but still below the stated upper reference. - Use equivalent fractions to verify the comparison exactly.

Solution

1. Choose $\frac{3}{4}$. 2. Rewrite with denominator $8$: $\frac{3}{4}=\frac{6}{8}$. 3. Since $\frac{6}{8}>\frac{5}{8}$, the value satisfies the inequality.

Answer

One valid answer is $\frac{3}{4}$, because $\frac{3}{4}=\frac{6}{8}>\frac{5}{8}$.

5497026

A mystery inequality has boundary $4$. The value $4$ is not a solution, $4.1$ is a solution, and $3.9$ is not. Write the inequality.

Hints

- Compare the known solution and nonsolution with the common boundary. - Use the excluded boundary to choose a strict symbol.

Solution

1. The solution $4.1$ lies above the boundary. 2. The nonsolution $3.9$ lies below the boundary. 3. The boundary is excluded. 4. Therefore, the inequality is $x>4$.

Answer

$x>4$

5497036

A science display can use any whole-number height $h$ from the set $\{30,31,32,33,34,35\}$ inches. The condition is $h>32$. List the allowed heights and explain why the answer is finite even though $h>32$ usually has infinitely many solutions.

Hints

- Apply the inequality only to the values the problem permits. - Distinguish an unrestricted solution set from a solution subset chosen from a list.

Solution

1. From the specified set, the values greater than $32$ are $33$, $34$, and $35$. 2. The unrestricted inequality has infinitely many solutions. 3. The answer is finite because only values from the given six-number set are being considered.

Answer

Allowed heights: $33\,\text{in.}$, $34\,\text{in.}$, and $35\,\text{in.}$. The specified candidate set limits the solutions to finitely many values.

5497046

Morgan tests $x=2.49$ in $x<2.5$ and says it fails because both numbers round to $2.5$ to the nearest tenth. Evaluate Morgan’s reasoning.

Hints

- Use the exact numbers given in the inequality test. - Consider whether rounding can change a strict comparison near the boundary.

Solution

1. Compare the original values, not rounded values. 2. Since $2.49<2.5$, the inequality is true. 3. Morgan’s reasoning is incorrect because rounding erased the small difference.

Answer

Morgan is incorrect. $2.49$ is a solution because $2.49<2.5$.

5540846

Leo says the solution set of $x<3$ should have a filled endpoint at $3$ because $3$ is the boundary, and it should extend toward smaller numbers. Identify which part of Leo's description is correct and which part is wrong. State the endpoint type and direction for a correct number-line representation.

Hints

- Decide first whether $3$ itself satisfies $x<3$. - A strict inequality treats its boundary differently from an inclusive inequality. - Then use the meaning of “less than” to determine the direction.

Solution

1. The inequality $x<3$ contains numbers smaller than $3$, so extending toward smaller numbers is correct. 2. The boundary value $3$ is not included because the inequality is strict. 3. Therefore, the endpoint at $3$ must be open, and the solution set extends to the left.

Answer

Use an open endpoint at $3$ and extend the solution set to the left. Leo has the direction correct but the endpoint type wrong.

5497016

Jada says $99.9$ is the greatest decimal solution to $x<100$. Is she correct? Give a greater decimal solution and explain why this inequality has no greatest decimal solution.

Hints

- Look for a decimal between $99.9$ and $100$. - Ask whether the same construction can be repeated with the new decimal.

Solution

1. Jada is not correct because $99.99$ is greater than $99.9$ and still less than $100$. 2. The same idea can be repeated: $99.999$ is greater than $99.99$ and still less than $100$. 3. Another decimal can always be placed closer to $100$ without reaching it, so there is no greatest decimal solution.

Answer

Jada is not correct. For example, $99.99$ is a greater solution. A decimal can always be chosen closer to $100$, so there is no greatest decimal solution.

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One-step inequalities and solution sets

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