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

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Example Questions

Example Question #21 : Equations / Inequalities

Solve: $\frac{4}{7}x > 7$ $\displaystyle \frac{4}{7}x > 7$

Possible Answers:

$x>\frac{1}{4}$ $\displaystyle x>\frac{1}{4}$

$x<\frac{49}{4}$ $\displaystyle x< \frac{49}{4}$

$x<\frac{45}{4}$ $\displaystyle x< \frac{45}{4}$

$x>\frac{49}{4}$ $\displaystyle x>\frac{49}{4}$

x>4 $\displaystyle x>4$

Correct answer:

$x>\frac{49}{4}$ $\displaystyle x>\frac{49}{4}$

Explanation:

To isolate the x variable, multiply the reciprocal of the fraction in front of the variable on both sides of the inequality.

$\frac{4}{7}x \cdot \frac{7}{4}> 7\cdot \frac{7}{4}$ $\displaystyle \frac{4}{7}x \cdot \frac{7}{4}> 7\cdot \frac{7}{4}$

Simplify both sides of the inequality.

The answer is: $x>\frac{49}{4}$ $\displaystyle x>\frac{49}{4}$

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Example Question #21 : Systems Of Inequalities

Solve the following inequality: $\frac{1}{3}x + \frac{2}{3}>3$ $\displaystyle \frac{1}{3}x + \frac{2}{3}>3$

Possible Answers:

x>7 $\displaystyle x>7$

$x>-\frac{7}{3}$ $\displaystyle x>-\frac{7}{3}$

x>1 $\displaystyle x>1$

x<7 $\displaystyle x< 7$

$x>\frac{7}{3}$ $\displaystyle x>\frac{7}{3}$

Correct answer:

x>7 $\displaystyle x>7$

Explanation:

In order to eliminate the fractions, multiply the inequality by three on both sides.

This will make the inequality easier to solve.

$3(\frac{1}{3}x + \frac{2}{3})>3(3)$ $\displaystyle 3(\frac{1}{3}x + \frac{2}{3})>3(3)$

Distribute and simplify both sides.

x+2>9 $\displaystyle x+2>9$

Subtract two from both sides.

$\displaystyle x+2-2>9-2$

The answer is: x>7 $\displaystyle x>7$

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Example Question #23 : Equations / Inequalities

Solve the inequality: $\frac{2}{3}x >15$ $\displaystyle \frac{2}{3}x >15$

Possible Answers:

$x>\frac{13}{3}$ $\displaystyle x>\frac{13}{3}$

$x<\frac{45}{2}$ $\displaystyle x< \frac{45}{2}$

$x>\frac{45}{2}$ $\displaystyle x>\frac{45}{2}$

$x>\frac{43}{3}$ $\displaystyle x>\frac{43}{3}$

$x<\frac{43}{3}$ $\displaystyle x< \frac{43}{3}$

Correct answer:

$x>\frac{45}{2}$ $\displaystyle x>\frac{45}{2}$

Explanation:

Multiply both sides by three halves.

$\frac{2}{3}x \cdot \frac{3}{2} >15 \cdot \frac{3}{2}$ $\displaystyle \frac{2}{3}x \cdot \frac{3}{2} >15 \cdot \frac{3}{2}$

Simplify both sides. Multiply the integer with the numerator on the right side of the inequality.

The answer is: $x>\frac{45}{2}$ $\displaystyle x>\frac{45}{2}$

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Example Question #22 : Systems Of Inequalities

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

Possible Answers:

$x<\frac{49}{2}$ $\displaystyle x< \frac{49}{2}$

$x>\frac{49}{2}$ $\displaystyle x>\frac{49}{2}$

$x\leq\frac{47}{7}$ $\displaystyle x\leq\frac{47}{7}$

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

$x<\frac{47}{7}$ $\displaystyle x< \frac{47}{7}$

Correct answer:

$x<\frac{49}{2}$ $\displaystyle x< \frac{49}{2}$

Explanation:

Solve this inequality by multiplying both sides by the reciprocal of the coefficient in front of the x-variable.

$\frac{2}{7}x \cdot \frac{7}{2} <7\cdot \frac{7}{2}$ $\displaystyle \frac{2}{7}x \cdot \frac{7}{2} < 7\cdot \frac{7}{2}$

Simplify both sides. Multiply the integer by the numerator on the right side of the inequality.

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

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Example Question #21 : Equations / Inequalities

Solve the following inequality: $\frac{2}{5}x>40$ $\displaystyle \frac{2}{5}x>40$

Possible Answers:

x>100 $\displaystyle x>100$

$x>\frac{35}{2}$ $\displaystyle x>\frac{35}{2}$

x>10 $\displaystyle x>10$

x>4 $\displaystyle x>4$

x>16 $\displaystyle x>16$

Correct answer:

x>100 $\displaystyle x>100$

Explanation:

Multiply the reciprocal of the fraction in front of the $\displaystyle x$ variable on both sides of the equation.

$\frac{2}{5}x \cdot \frac{5}{2}>40\cdot \frac{5}{2}$ $\displaystyle \frac{2}{5}x \cdot \frac{5}{2}>40\cdot \frac{5}{2}$

Simplify both sides.

$x>\frac{200}{2}$ $\displaystyle x>\frac{200}{2}$

Reduce the fraction on the right side.

The answer is: x>100 $\displaystyle x>100$

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Example Question #21 : Systems Of Inequalities

Solve the inequality: $\frac{3}{4}x>-4$ $\displaystyle \frac{3}{4}x>-4$

Possible Answers:

$x<-\frac{16}{3}$ $\displaystyle x< -\frac{16}{3}$

x>-2 $\displaystyle x>-2$

$x>-\frac{1}{3}$ $\displaystyle x>-\frac{1}{3}$

x>-4 $\displaystyle x>-4$

$x>-\frac{16}{3}$ $\displaystyle x>-\frac{16}{3}$

Correct answer:

$x>-\frac{16}{3}$ $\displaystyle x>-\frac{16}{3}$

Explanation:

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

$\frac{3}{4}x\cdot \frac{4}{3}>-4\cdot \frac{4}{3}$ $\displaystyle \frac{3}{4}x\cdot \frac{4}{3}>-4\cdot \frac{4}{3}$

Simplify both sides. There is no need to switch the sign. Multiply the negative integer with the numerator on the right side of the inequality.

The answer is: $x>-\frac{16}{3}$ $\displaystyle x>-\frac{16}{3}$

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Example Question #21 : Systems Of Inequalities

Solve the inequality: $9x\leq -\frac{1}{3}$ $\displaystyle 9x\leq -\frac{1}{3}$

Possible Answers:

$x< -\frac{1}{27}$ $\displaystyle x< -\frac{1}{27}$

$x\leq -\frac{1}{27}$ $\displaystyle x\leq -\frac{1}{27}$

$x\leq -\frac{1}{3}$ $\displaystyle x\leq -\frac{1}{3}$

$x\geq -\frac{1}{27}$ $\displaystyle x\geq -\frac{1}{27}$

$x\geq -\frac{1}{3}$ $\displaystyle x\geq -\frac{1}{3}$

Correct answer:

$x\leq -\frac{1}{27}$ $\displaystyle x\leq -\frac{1}{27}$

Explanation:

In order to isolate the x-variable, we will need to divide both sides by nine. To avoid writing a complex fraction on the right side, dividing by nine is similar to multiplying by one-ninth.

Multiply by one-ninth on both sides.

$9x \cdot \frac{1}{9}\leq -\frac{1}{3} \cdot \frac{1}{9}$ $\displaystyle 9x \cdot \frac{1}{9}\leq -\frac{1}{3} \cdot \frac{1}{9}$

Simplify both sides. Multiply the denominator with denominator on the right side. Since we have divide by a positive number on both sides, there is no need to switch the sign.

The answer is: $x\leq -\frac{1}{27}$ $\displaystyle x\leq -\frac{1}{27}$

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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 #21 : Equations / Inequalities

Example Question #21 : Systems Of Inequalities

Example Question #23 : Equations / Inequalities

Example Question #22 : Systems Of Inequalities

Example Question #21 : Equations / Inequalities

Example Question #21 : Systems Of Inequalities

Example Question #21 : Systems Of Inequalities

All Algebra 1 Resources

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