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

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Example Questions

Example Question #83 : Equations / Inequalities

Solve for $\displaystyle x$:

$\displaystyle 3x-1>x+7$

Possible Answers:

x>2 $\displaystyle x>2$

x<4 $\displaystyle x< 4$

None of the other answers

x<2 $\displaystyle x< 2$

x>4 $\displaystyle x>4$

Correct answer:

x>4 $\displaystyle x>4$

Explanation:

To solve for $\displaystyle x$, separate the integers and $\displaystyle x$'s by adding 1 and subtracting $\displaystyle x$ from both sides to get 2x>8 $\displaystyle 2x>8$. Then, divide both sides by 2 to get x>4 $\displaystyle x>4$. Since you didn't divide by a negative number, the sign does not need to be reversed.

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Example Question #2 : How To Find The Solution To An Inequality With Division

Solve the following: -2x+3>4 $\displaystyle -2x+3>4$

Possible Answers:

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

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

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

$x>-\frac{7}{2}$ $\displaystyle x>-\frac{7}{2}$

Correct answer:

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

Explanation:

$\displaystyle -2x+3-3>4-3$

-2x>1 $\displaystyle -2x>1$

$\frac{-2x}{-2}<\frac{1}{-2}$ $\displaystyle \frac{-2x}{-2}< \frac{1}{-2}$ Don't forget to change the direction of the inequality sign when dividing by a negative number!

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

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Example Question #3 : How To Find The Solution To An Inequality With Division

Give the solution set of the inequality:

$\displaystyle -4x + 17 > 81$

Possible Answers:

$\left (-16, \infty \right )$ $\displaystyle \left (-16, \infty \right )$

$\displaystyle (-16,16)$

The set of all real numbers

$(-\infty ,-16)$ $\displaystyle (-\infty ,-16)$

$(-\infty ,16)$ $\displaystyle (-\infty ,16)$

Correct answer:

$(-\infty ,-16)$ $\displaystyle (-\infty ,-16)$

Explanation:

$\displaystyle -4x + 17 > 81$

$\displaystyle -4x + 17 -17 > 81 -17$

$\displaystyle -4x > 64$

$-4x \div (-4)< 64 \div (-4)$ $\displaystyle -4x \div (-4)< 64 \div (-4)$

Note change in direction of the inequality symbol when the expressions are divided by a negative number.

x < -16 $\displaystyle x < -16$

or, in interval form,

$(-\infty ,-16)$ $\displaystyle (-\infty ,-16)$

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Example Question #4 : How To Find The Solution To An Inequality With Division

Give the solution set of the inequality:

$-5x + 17 \geq 82$ $\displaystyle -5x + 17 \geq 82$

Possible Answers:

$\left [13, \infty \right )$ $\displaystyle \left [13, \infty \right )$

$\displaystyle (-13,13)$

The inequality has no solution.

$[-13,\infty)$ $\displaystyle [-13,\infty)$

$(-\infty , -13]$ $\displaystyle (-\infty , -13]$

Correct answer:

$(-\infty , -13]$ $\displaystyle (-\infty , -13]$

Explanation:

$-5x + 17 \geq 82$ $\displaystyle -5x + 17 \geq 82$

$-5x + 17 -17 \geq 82-17$ $\displaystyle -5x + 17 -17 \geq 82-17$

$-5x \geq 65$ $\displaystyle -5x \geq 65$

$-5x \div (-5) \leq 65\div (-5)$ $\displaystyle -5x \div (-5) \leq 65\div (-5)$

Note change in direction of the inequality symbol when the expressions are divided by a negative number.

$x \leq -13$ $\displaystyle x \leq -13$

or, in interval form,

$(-\infty , -13]$ $\displaystyle (-\infty , -13]$

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Example Question #5 : How To Find The Solution To An Inequality With Division

Give the solution set of the inequality:

$-5x + 27 \leq 62$ $\displaystyle -5x + 27 \leq 62$

Possible Answers:

$[-7,\infty )$ $\displaystyle [-7,\infty )$

[-7,7] $\displaystyle [-7,7]$

$[7,\infty )$ $\displaystyle [7,\infty )$

The inequality has no solution.

$(-\infty , -7]$ $\displaystyle (-\infty , -7]$

Correct answer:

$[-7,\infty )$ $\displaystyle [-7,\infty )$

Explanation:

$-5x + 27 \leq 62$ $\displaystyle -5x + 27 \leq 62$

$-5x + 27 -27 \leq 62-27$ $\displaystyle -5x + 27 -27 \leq 62-27$

$-5x \leq 35$ $\displaystyle -5x \leq 35$

$-5x \div (-5) \leq 35\div (-5)$ $\displaystyle -5x \div (-5) \leq 35\div (-5)$

Note change in direction of the inequality symbol when the expressions are divided by a negative number.

$x \geq -7$ $\displaystyle x \geq -7$

or, in interval form,

$[-7,\infty )$ $\displaystyle [-7,\infty )$

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Example Question #91 : Equations / Inequalities

Give the solution set of the inequality:

$\displaystyle -4x - 11 < 57$

Possible Answers:

$(-\infty,-17 )$ $\displaystyle (-\infty,-17 )$

$(-17,\infty )$ $\displaystyle (-17,\infty )$

The set of all real numbers

$\displaystyle (-17,17)$

$(17,\infty )$ $\displaystyle (17,\infty )$

Correct answer:

$(-17,\infty )$ $\displaystyle (-17,\infty )$

Explanation:

$\displaystyle -4x - 11 < 57$

$\displaystyle -4x - 11 + 11 < 57 + 11$

$\displaystyle -4x < 68$

$-4x \div (-4) > 68 \div (-4)$ $\displaystyle -4x \div (-4) > 68 \div (-4)$

Note change in direction of the inequality symbol when the expressions are divided by a negative number.

x > -17 $\displaystyle x > -17$

or, in interval form,

$(-17,\infty )$ $\displaystyle (-17,\infty )$

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Example Question #92 : Equations / Inequalities

Give the solution set of the inequality:

$\displaystyle -4x + 11 < 67$

Possible Answers:

$(-\infty ,14 )$ $\displaystyle (-\infty ,14 )$

$\displaystyle (-14,14)$

The set of all real numbers

$(-14, \infty )$ $\displaystyle (-14, \infty )$

$(-\infty ,-14 )$ $\displaystyle (-\infty ,-14 )$

Correct answer:

$(-14, \infty )$ $\displaystyle (-14, \infty )$

Explanation:

$\displaystyle -4x + 11 < 67$

$\displaystyle -4x + 11 - 11 < 67- 11$

$\displaystyle -4x < 56$

$-4x \div (-4) > 56 \div (-4)$ $\displaystyle -4x \div (-4) > 56 \div (-4)$

Note change in direction of the inequality symbol when the expressions are divided by a negative number.

x > -14 $\displaystyle x > -14$

or, in interval form,

$(-14, \infty )$ $\displaystyle (-14, \infty )$

Report an Error

Example Question #93 : Equations / Inequalities

Solve for $\displaystyle x$:

$\displaystyle 2-6x>10-2x$

Possible Answers:

x>2 $\displaystyle x>2$

x<-2 $\displaystyle x< -2$

None of the other answers

x<2 $\displaystyle x< 2$

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

Correct answer:

x<-2 $\displaystyle x< -2$

Explanation:

First, add $\displaystyle 2x$ and subtract $\displaystyle 2$ from both sides of the inequality to get -4x>8 $\displaystyle -4x>8$.

Then, divide both sides by $\displaystyle -4$ and reverse the sign since you are dividing by a negative number.

This gives you x<-2 $\displaystyle x< -2$.

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Example Question #1 : How To Find The Solution To An Inequality With Division

Find the solution set to the following compound inequality statement:

$4y + 15 < 91 \textrm{ and } 7y -12 > 86$ $\displaystyle 4y + 15 < 91 \textrm{ and } 7y -12 > 86$

Possible Answers:

(14,19) $\displaystyle (14,19)$

$(-\infty , \infty )$ $\displaystyle (-\infty , \infty )$

$(14, \infty )$ $\displaystyle (14, \infty )$

$(-\infty , 19)$ $\displaystyle (-\infty , 19)$

$(-\infty , 14) \cup (19, \infty )$ $\displaystyle (-\infty , 14) \cup (19, \infty )$

Correct answer:

(14,19) $\displaystyle (14,19)$

Explanation:

Solve each of these two inequalities separately:

$\displaystyle 4y + 15 < 91$

$\displaystyle 4y + 15 - 15 < 91- 15$

4y < 76 $\displaystyle 4y < 76$

$4y \div 4 < 76 \div 4$ $\displaystyle 4y \div 4 < 76 \div 4$

y < 19 $\displaystyle y < 19$, or, in interval form, $(-\infty , 19)$ $\displaystyle (-\infty , 19)$

$\displaystyle 7y -12 > 86$

$\displaystyle 7y -12 + 12 > 86 + 12$

7y > 98 $\displaystyle 7y > 98$

$7y \div 7 > 98 \div 7$ $\displaystyle 7y \div 7 > 98 \div 7$

y > 14 $\displaystyle y > 14$, or, in interval form, $(14, \infty )$ $\displaystyle (14, \infty )$

The two inequalities are connected with an "and", so we take the intersection of the two intervals.

$(14, \infty ) \cap (-\infty , 19) = (14,19)$ $\displaystyle (14, \infty ) \cap (-\infty , 19) = (14,19)$

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Example Question #95 : Equations / Inequalities

Solve for $\displaystyle x$:

$0.4 x - 1.7 \geq -8.5$ $\displaystyle 0.4 x - 1.7 \geq -8.5$

Possible Answers:

$[1.7, \infty )$ $\displaystyle [1.7, \infty )$

$[-1.7, \infty )$ $\displaystyle [-1.7, \infty )$

$[17, \infty )$ $\displaystyle [17, \infty )$

$[-17, \infty )$ $\displaystyle [-17, \infty )$

The inequality has no solution.

Correct answer:

$[-17, \infty )$ $\displaystyle [-17, \infty )$

Explanation:

$0.4 x - 1.7 \geq -8.5$ $\displaystyle 0.4 x - 1.7 \geq -8.5$

$0.4 x - 1.7 + 1.7 \geq -8.5+ 1.7$ $\displaystyle 0.4 x - 1.7 + 1.7 \geq -8.5+ 1.7$

$0.4 x \geq -6.8$ $\displaystyle 0.4 x \geq -6.8$

$0.4 x \div 0.4 \geq -6.8\div 0.4$ $\displaystyle 0.4 x \div 0.4 \geq -6.8\div 0.4$

$x \geq -17$ $\displaystyle x \geq -17$

or, in interval form, $[-17, \infty )$ $\displaystyle [-17, \infty )$

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

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #83 : Equations / Inequalities

Example Question #2 : How To Find The Solution To An Inequality With Division

Example Question #3 : How To Find The Solution To An Inequality With Division

Example Question #4 : How To Find The Solution To An Inequality With Division

Example Question #5 : How To Find The Solution To An Inequality With Division

Example Question #91 : Equations / Inequalities

Example Question #92 : Equations / Inequalities

Example Question #93 : Equations / Inequalities

Example Question #1 : How To Find The Solution To An Inequality With Division

Example Question #95 : Equations / Inequalities

All Algebra 1 Resources

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