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Algebra 1 : Equations / Inequalities

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Example Questions

Example Question #81 : Systems Of Inequalities

Solve the inequality: 6<x-19 $\displaystyle 6< x-19$

Possible Answers:

x>-13 $\displaystyle x>-13$

x>-25 $\displaystyle x>-25$

x>25 $\displaystyle x>25$

x<-25 $\displaystyle x< -25$

x<25 $\displaystyle x< 25$

Correct answer:

x>25 $\displaystyle x>25$

Explanation:

Solve by adding nineteen on both sides.

6+19<x-19+19 $\displaystyle 6+19< x-19+19$

Simplify both sides.

25<x $\displaystyle 25< x$

This is also the same as: x>25 $\displaystyle x>25$

The answer is: x>25 $\displaystyle x>25$

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Example Question #81 : Equations / Inequalities

Solve the inequality: $z-3\geq 16$ $\displaystyle z-3\geq 16$

Possible Answers:

$z\leq 19$ $\displaystyle z\leq 19$

z<19 $\displaystyle z< 19$

$z\geq13$ $\displaystyle z\geq13$

$z\leq13$ $\displaystyle z\leq13$

$z\geq 19$ $\displaystyle z\geq 19$

Correct answer:

$z\geq 19$ $\displaystyle z\geq 19$

Explanation:

In order to solve this inequality, add three on both sides.

$z-3+3\geq 16+3$ $\displaystyle z-3+3\geq 16+3$

Simplify both sides of the inequality.

The answer is: $z\geq 19$ $\displaystyle z\geq 19$

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Example Question #83 : Systems Of Inequalities

Solve the following inequality: $-x-9\leq-6$ $\displaystyle -x-9\leq-6$

Possible Answers:

$x\leq -3$ $\displaystyle x\leq -3$

$x\leq 15$ $\displaystyle x\leq 15$

$x\geq 3$ $\displaystyle x\geq 3$

$x\leq 3$ $\displaystyle x\leq 3$

$x\geq -3$ $\displaystyle x\geq -3$

Correct answer:

$x\geq -3$ $\displaystyle x\geq -3$

Explanation:

Add six on both sides

$-x-9+6\leq-6+6$ $\displaystyle -x-9+6\leq-6+6$

Simplify both sides.

$-x-3\leq0$ $\displaystyle -x-3\leq0$

Add $\displaystyle x$ on both sides.

$-x-3+x\leq0+x$ $\displaystyle -x-3+x\leq0+x$

Simplify the left side. Since we are adding a negative variable, it is not necessary to change the sign.

$-3\leq x$ $\displaystyle -3\leq x$

The answer is: $x\geq -3$ $\displaystyle x\geq -3$

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Example Question #84 : Systems Of Inequalities

Solve the inequality: $\displaystyle -3-3x>-2x$

Possible Answers:

$x\geq-3$ $\displaystyle x\geq-3$

x<-3 $\displaystyle x< -3$

$x\leq-3$ $\displaystyle x\leq-3$

x>-3 $\displaystyle x>-3$

x<3 $\displaystyle x< 3$

Correct answer:

x<-3 $\displaystyle x< -3$

Explanation:

Group the x-variables by adding $\displaystyle 3x$ on both sides of the equation.

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

Simplify both sides of the equation.

-3>x $\displaystyle -3>x$

Since we did not divide by a negative number, we do not need to switch the direction of the sign.

The answer is: x<-3 $\displaystyle x< -3$

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Example Question #82 : Equations / Inequalities

Solve the inequality: x-15<66 $\displaystyle x-15< 66$

Possible Answers:

x<51 $\displaystyle x< 51$

x<-81 $\displaystyle x< -81$

x>-81 $\displaystyle x>-81$

x<-51 $\displaystyle x< -51$

x<81 $\displaystyle x< 81$

Correct answer:

x<81 $\displaystyle x< 81$

Explanation:

In order to isolate the x-variable, add 15 on both sides of equation.

x-15+15<66+15 $\displaystyle x-15+15< 66+15$

Simplify both sides of the inequality.

The answer is: x<81 $\displaystyle x< 81$

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

Solve for $\displaystyle x$:

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

Possible Answers:

x<4 $\displaystyle x< 4$

x>2 $\displaystyle x>2$

x<2 $\displaystyle x< 2$

x>4 $\displaystyle x>4$

None of the other answers

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:

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

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

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

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

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:

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

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

$\displaystyle (-16,16)$

The set of all real numbers

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

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:

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

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

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

The inequality has no solution.

$\displaystyle (-13,13)$

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 #83 : Equations / Inequalities

Give the solution set of the inequality:

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

Possible Answers:

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

The inequality has no solution.

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

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

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

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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Algebra 1 : Equations / Inequalities

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #81 : Systems Of Inequalities

Example Question #81 : Equations / Inequalities

Example Question #83 : Systems Of Inequalities

Example Question #84 : Systems Of Inequalities

Example Question #82 : Equations / Inequalities

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

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 #83 : Equations / Inequalities

All Algebra 1 Resources

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