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

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Example Questions

Example Question #1 : Equations / Inequalities

Solve the inequality:

$\small \frac{5}{7}x > 15$

Possible Answers:

$\small x>\frac{25}{49}$

$\small x<-21$

$\small x> \frac{110}{7}$

$\small x>21$

$\small x> \frac{100}{7}$

Correct answer:

$\small x>21$

Explanation:

In order to isolate the variable, we need to remove the coefficient. Since the operation between $\tiny \small \frac{5}{7}$ and $\small x$ is multiplication, we may want to divide both sides of the inequality by $\tiny \small \frac{5}{7}$ . Although this is a valid step, in order to simplify matters, instead of dividing by $\tiny \small \frac{5}{7}$ , we can multiplying by $\small \frac{7}{5}$ .

(Note: dividing by $\frac{5}{7}$ is exactly the same as multplying by the reciprocal, $\frac{7}{5}$ .)

Thus,

$\small x> \frac{15}{1} \cdot\frac{7}{5}$ and after multiplying and simplifying, we obtain $\small x>21$ .

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Example Question #2 : Systems Of Inequalities

Solve for :

3x-2-x>-2x+6

Possible Answers:

x>2

None of the other answers

x<2

x<-2

x>-2

Correct answer:

x>2

Explanation:

Simplify 3x-2-x>-2x+6 by combining like terms to get 2x-2>-2x+6 .

Then, add and to both sides to separate the 's and intergers. This gives you 4x>8 .

Divide both sides to get x>2 . Since we didn't divide by a negative number, there is no need to reverse the sign.

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Example Question #3 : Systems Of Inequalities

Solve for :

$\frac{7}{9} x > \frac{5}{3}$

Possible Answers:

$\left ( \frac{13}{7},\infty \right )$

$\left ( \frac{5}{3},\infty \right )$

$\left ( \frac{7}{3},\infty \right )$

$\left ( \frac{11}{7},\infty \right )$

$\left ( \frac{15}{7},\infty \right )$

Correct answer:

$\left ( \frac{15}{7},\infty \right )$

Explanation:

$\frac{7}{9} x > \frac{5}{3}$

$\frac{7}{9} x \div \frac{7}{9}> \frac{5}{3} \div \frac{7}{9}$

$\frac{7}{9} x \cdot \frac{9}{7}> \frac{5}{3}\cdot \frac{9}{7}$

Cross-cancel:

$x> \frac{5}{1}\cdot \frac{3}{7}$

$x> \frac{15}{7}$

or, in interval form, $\left ( \frac{15}{7},\infty \right )$ .

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Example Question #4 : Systems Of Inequalities

Solve for :

$\frac{2}{9} x + \frac{1}{3} < \frac{5}{6}$

Possible Answers:

The inequality has no solution.

$x < \frac{21}{4}$

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

$x < \frac{13}{4}$

$x < \frac{17}{4}$

Correct answer:

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

Explanation:

Eliminate fractions by multiplying by the least common denominator - LCM(3,6,9) = 18 .

$18 \cdot \left (\frac{2}{9} x + \frac{1}{3} \right ) <18 \cdot \frac{5}{6}$

$\frac{18}{1} \cdot \frac{2}{9} x + \frac{18}{1} \cdot \frac{1}{3} \right ) <\frac{18}{1} \cdot \frac{5}{6}$

Cross-cancel:

$\frac{2}{1} \cdot \frac{2}{1} x + \frac{6}{1} \cdot \frac{1}{1} \right ) <\frac{3}{1} \cdot \frac{5}{1}$

4x + 6 < 15

4x + 6 - 6 < 15 - 6

4x < 9

$4x \div 4 < 9 \div 4$

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

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Example Question #5 : Systems Of Inequalities

Find the solution set for :

$-\frac{7}{10} < -\frac{2}{5} x \leq \frac{1}{10}$

Possible Answers:

$\left ( - \frac{1}{4},\frac{7}{4} \right ]$

$\left [ -\frac{7}{4},-\frac{1}{4} \right )$

$\left [ - \frac{1}{4},\frac{7}{4} \right )$

$\left ( - \frac{7}{4},\frac{1}{4} \right ]$

$\left [ \frac{1}{4},\frac{7}{4} \right )$

Correct answer:

$\left [ - \frac{1}{4},\frac{7}{4} \right )$

Explanation:

$-\frac{7}{10} < -\frac{2}{5} x \leq \frac{1}{10}$

$- \frac{5}{2} \cdot \left (-\frac{7}{10} \right )> - \frac{5}{2} \cdot \left ( -\frac{2}{5}x \right ) \geq - \frac{5}{2} \cdot \frac{1}{10}$

Note the switch in inequality symbols when the numbers are multiplied by a negative number.

$\frac{5}{2} \cdot \frac{7}{10} > x \geq - \frac{5}{2} \cdot \frac{1}{10}$

Cross-cancel:

$\frac{1}{2} \cdot \frac{7}{2} > x \geq - \frac{1}{2} \cdot \frac{1}{2}$

$\frac{7}{4} > x \geq - \frac{1}{4}$

or, in interval form, $\left [ - \frac{1}{4},\frac{7}{4} \right )$

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Example Question #1 : Systems Of Inequalities

Solve for :

3x+4>9x-8

Possible Answers:

x>2

x<2

x<-2

x>-2

None of the other answers are correct.

Correct answer:

x<2

Explanation:

Subtract 4 from both sides. Then subtract 9x:

-6x>-12

Next divide both sides by -6. Don't forget to switch the inequality because of the negative sign!

x<2

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Example Question #2 : Equations / Inequalities

Solve for :

3x-12>x+2

Possible Answers:

x<7

x<5

None of the other answers are correct.

x>7

x>5

Correct answer:

x>7

Explanation:

To solve the inequality, subtract and add 12 to both sides to separate the from the integers:

2x>14

Divide both sides by 2:

x>7

Note: The inequality sign is only flipped when dividing by negative numbers.

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Example Question #2131 : Algebra 1

Solve for :

3x-5>5x+1+x

Possible Answers:

x<-2

x>-2

x>4

None of the other answers are correct.

x<4

Correct answer:

x<-2

Explanation:

First, combine the like terms on the righthand side of the inequality to get 3x-5>6x+1 .

Then, subtract and from both sides to get -6>3x .

Finally, divide both sides by : x<-2

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Example Question #2 : Systems Of Inequalities

Solve this inequality.

-3x+2 < x+4

Possible Answers:

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

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

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

x>2

x<1

Correct answer:

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

Explanation:

Isolate all the terms with on one side and the other terms on the other side and solve for .

-3x+2 < x+4

First subtract the an x from both sides of the inequality. The subtract 2 from each side. This results in the following inequality.

-4x<2

Here, we need to divide both sides by . However, whenever we divide or multiply and inequality by a negative number, we have to also change the direction of the inequality.

The final answer becomes

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

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Example Question #2 : Equations / Inequalities

$2x\leq6$

Find all of the solutions to this inequality.

Possible Answers:

$(-\infty,3]$

$(3,\infty )$

$[6,\infty )$

$(\infty, 6)$

$[\infty ,3]$

Correct answer:

$(-\infty,3]$

Explanation:

To solve an inequality, isolate the variable on one side with all other constants on the other side. To accomplish this, perform opposite operations to manipulate the inequality.

First, isolate the x by multiplying each side by two.

Whatever you do to one side you must also do to the other side.

$\frac{2x}{{\color{Blue} 2}}\leq\frac{6}{{\color{Blue} 2}}$

This gives you:

$x\leq3$

The answer, therefore, is $(-\infty ,3]$ .

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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 #1 : Equations / Inequalities

Example Question #2 : Systems Of Inequalities

Example Question #3 : Systems Of Inequalities

Example Question #4 : Systems Of Inequalities

Example Question #5 : Systems Of Inequalities

Example Question #1 : Systems Of Inequalities

Example Question #2 : Equations / Inequalities

Example Question #2131 : Algebra 1

Example Question #2 : Systems Of Inequalities

Example Question #2 : Equations / Inequalities

All Algebra 1 Resources

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