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Algebra 1 : How to find the solution to an inequality with multiplication

Study concepts, example questions & explanations for Algebra 1

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Example Questions

Example Question #11 : Equations / Inequalities

Solve the following inequality: $\frac{x}{4}\leq4$

Possible Answers:

$x\leq 16$

$x\geq 1$

$x\leq \frac{1}{16}$

$x\leq 1$

$x\geq 16$

Correct answer:

$x\leq 16$

Explanation:

In order to isolate variable, multiply by four on both sides of the equation.

$\frac{x}{4}\cdot 4\leq4 \cdot 4$

Simplify both sides of the inequality.

$x\leq 16$

The answer is: $x\leq 16$

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Example Question #11 : Equations / Inequalities

Solve: $\frac{x}{7}>7$

Possible Answers:

x>14

x<14

x<49

x>49

x>1

Correct answer:

x>49

Explanation:

Multiply by seven on both sides of the equation. There is no need to change the direction of the sign unless there is a negative sign.

$\frac{x}{7} \cdot 7>7\cdot 7$

Simplify both sides.

The answer is: x>49

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Example Question #13 : Equations / Inequalities

Solve the inequality: $\frac{6}{7}x >42$

Possible Answers:

x>1

x<36

x>36

x>49

x<49

Correct answer:

x>49

Explanation:

In order to isolate the unknown variable, we will need to multiply the reciprocal of the coefficient on both sides of the inequality sign.

$\frac{6}{7}x \cdot \frac{7}{6}>42\cdot \frac{7}{6}$

Simplify both sides of the equation.

$x>42\cdot \frac{7}{6}$

The answer is: x>49

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Example Question #14 : Equations / Inequalities

Solve: $\frac{9x}{7}<3$

Possible Answers:

$x<\frac{7}{3}$

$x<\frac{12}{7}$

$x<\frac{27}{7}$

$x\leq\frac{7}{3}$

$x<\frac{9}{7}$

Correct answer:

$x<\frac{7}{3}$

Explanation:

Multiply both sides by the reciprocal of the fraction in front of the variable.

$\frac{9x}{7} \times \frac{7}{9}<3 \times \frac{7}{9}$

Simplify both sides. The nine on the right side of the inequality can be split into factors.

$x<\frac{3}{1} \times \frac{7}{3\times 3}$

The answer is: $x<\frac{7}{3}$

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Example Question #2142 : Algebra 1

Solve the inequality and rewrite in interval notation: $\frac{3}{4}x \geq \frac{1}{3}$

Possible Answers:

$(\frac{4}{9},\infty)$

$[\frac{1}{4},\infty)$

$(\frac{1}{4},\infty)$

$[\frac{4}{9},\infty)$

$[\frac{4}{9},\infty]$

Correct answer:

$[\frac{4}{9},\infty)$

Explanation:

In order to isolate the variable, we can multiply both sides by the reciprocal of the coefficient in front of the x variable.

$\frac{3}{4}x \cdot \frac{4}{3} \geq \frac{1}{3} \cdot \frac{4}{3}$

Simplify both sides of the equation.

$x\geq \frac{4}{9}$

This indicates that the x-variable is four ninths or greater. Use a bracket sign to indicate that it includes the fraction. The infinity isn't finite, which means that a parenthesis should enclose this symbol instead of a bracket.

The answer is: $[\frac{4}{9},\infty)$

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Example Question #13 : Equations / Inequalities

Solve the inequality: $-\frac{x}{6}>6$

Possible Answers:

x<-1

x<-36

x>-36

$x\leq -36$

x>-1

Correct answer:

x<-36

Explanation:

In order to isolate the x variable, we will need to divide by negative one sixth on both sides. The result will switch the sign of the inequality.

This is also the same as multiplying by negative six on both sides.

$-\frac{x}{6} \cdot -6>6 \cdot -6$

x>-36

Switch the sign.

x<-36

Upon testing values that are less than negative 36, we will find that those values will satisfy the inequality instead of x>-36 .

The answer is: x<-36

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Example Question #14 : Equations / Inequalities

Solve the following inequality: $\frac{2x}{7}\geq7$

Possible Answers:

$x\leq2$

$x\geq\frac{1}{2}$

$x\geq\frac{49}{2}$

$x\leq\frac{49}{2}$

$x\leq\frac{1}{2}$

Correct answer:

$x\geq\frac{49}{2}$

Explanation:

In order to isolate the x-variable, we will need to multiply both sides by the reciprocal of the coefficient in front of the x.

$\frac{2x}{7}\cdot \frac{7}{2}\geq7 \cdot \frac{7}{2}$

Simplify both sides. A whole number will multiply to the numerator of a fraction.

The answer is: $x\geq\frac{49}{2}$

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Example Question #12 : Equations / Inequalities

Solve: $\frac{1}{8}x > -4$

Possible Answers:

x>32

x<-32

$x>-\frac{1}{2}$

x>-32

x<32

Correct answer:

x>-32

Explanation:

To isolate the x-variable, we will need to multiply by eight on both sides.

$\frac{1}{8}x \cdot 8> -4\cdot 8$

Simplify both sides of the equation.

The answer is: x>-32

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Example Question #16 : Equations / Inequalities

Find the solution to the inequality: $\frac{x}{9}<\frac{1}{18}$

Possible Answers:

$x\geq2$

$x<\frac{1}{2}$

$x\leq2$

$x>-\frac{1}{2}$

x<2

Correct answer:

$x<\frac{1}{2}$

Explanation:

In order to isolate the x-variable, we will need to multiply by nine on both sides of the inequality.

$\frac{x}{9} \cdot 9<\frac{1}{18}\cdot 9$

Simplify both sides of the equation.

$x<\frac{9}{18}$

Reduce the fraction on the right side.

The answer is: $x<\frac{1}{2}$

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Example Question #2143 : Algebra 1

Solve the inequality: $-\frac{2}{3}x< 5$

Possible Answers:

$x>-\frac{10}{3}$

$x<\frac{15}{2}$

$x<-\frac{15}{2}$

$x>\frac{15}{2}$

$x>-\frac{15}{2}$

Correct answer:

$x>-\frac{15}{2}$

Explanation:

In order to isolate the x-variable, we will need to multiply by the negative reciprocal of two-thirds on both sides of the equation.

$-\frac{2}{3}x \cdot -\frac{3}{2}< 5 \cdot-\frac{3}{2}$

Simplify both sides of the inequality. Since we have a negative coefficient, and multiplying by the reciprocal is similar to dividing, we will need to switch the sign.

$x>-\frac{15}{2}$

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Algebra 1 : How to find the solution to an inequality with multiplication

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #11 : Equations / Inequalities

Example Question #11 : Equations / Inequalities

Example Question #13 : Equations / Inequalities

Example Question #14 : Equations / Inequalities

Example Question #2142 : Algebra 1

Example Question #13 : Equations / Inequalities

Example Question #14 : Equations / Inequalities

Example Question #12 : Equations / Inequalities

Example Question #16 : Equations / Inequalities

Example Question #2143 : Algebra 1

All Algebra 1 Resources

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