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Algebra II : Solving Inequalities

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Example Question #101 : Inequalities

Find the solution space for the inequality below:

5-4x>17

Possible Answers:

x<-3

x>-3

x>3

x<3

Correct answer:

x<-3

Explanation:

We begin solving the inequality the same way that we would solve an equality, we work to get x by itself.

5-4x>17 (subtract 5 from both sides)

-4x>12 (divide both sides by (-4) and FLIP THE INEQUALITY)

x<-3 (this is the solution space)

The key to solving this problem which is easy to forget is to FLIP THE INEQUALITY SYMBOL.

The reason we did this is because the sign changed on both sides.

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Example Question #102 : Inequalities

Solve the inequality: $4x-6\geq11x+4$

Possible Answers:

$x\geq -\frac{7}{10}$

$x\leq -\frac{2}{3}$

$x\geq -\frac{2}{3}$

$x\geq -\frac{10}{7}$

$x\leq -\frac{10}{7}$

Correct answer:

$x\leq -\frac{10}{7}$

Explanation:

Subtract on both sides.

$4x-6-4x\geq11x+4-4x$

$-6\geq7x+4$

Subtract four on both sides.

$-6-4\geq7x+4-4$

Simplify both sides.

$-10\geq7x$

Divide by seven on both sides.

$\frac{-10}{7}\geq\frac{7x}{7}$

$-\frac{10}{7} \geq x$

The answer is: $x\leq -\frac{10}{7}$

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Example Question #103 : Inequalities

Solve the inequality: 5(-3x+2)>-5x

Possible Answers:

x>-1

x<-1

x<1

x>1

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

Correct answer:

x<1

Explanation:

Distribute the five through the binomial.

5(-3x)+5(2)>-5x

Simplify the inequality.

-15x+10>-5x

Add 15x on both sides.

-15x+10+15x>-5x+15x

10>10x

Divide by ten on both sides.

1>x

The answer is: x<1

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Example Question #104 : Inequalities

Solve the inequality: 5(3x-3)>8x

Possible Answers:

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

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

$x<\frac{23}{15}$

$x>\frac{23}{15}$

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

Correct answer:

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

Explanation:

Simplify the left side of 5(3x-3)>8x using distribution.

15x-15>8x

Subtract on both sides.

15x-15-8x>8x-8x

7x-15>0

Add 15 on both sides.

7x-15+15>0+15

7x>15

Divide by seven on both sides.

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

The answer is: $x>\frac{15}{7}$

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Example Question #105 : Inequalities

Solve the inequality: $3(3-4x)\leq 26x-2$

Possible Answers:

$x\geq -\frac{11}{38}$

$x\geq -\frac{7}{38}$

$x\geq \frac{11}{38}$

$x\geq \frac{11}{14}$

$x\geq -\frac{11}{14}$

Correct answer:

$x\geq \frac{11}{38}$

Explanation:

Simplify the left side of the inequality. Distribute the three across both terms in the binomial.

$3(3-4x)\leq 26x-2$

$9-12x\leq 26x-2$

Add 12x on both sides.

$9-12x+12\leq 26x-2+12x$

$9\leq 38x-2$

Add 2 on both sides of the equation.

$9+2\leq 38x-2+2$

$11\leq 38x$

Divide by 38 on both sides.

$\frac{11}{38}\leq \frac{38x}{38}$

$\frac{11}{38}\leq x$

The answer is: $x\geq \frac{11}{38}$

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Example Question #106 : Inequalities

Solve the inequality: -3x+26>14-7x

Possible Answers:

x>3

x>-3

x>-4

x<-3

x<-4

Correct answer:

x>-3

Explanation:

Add on both sides.

-3x+26+(7x)>14-7x+(7x)

Simplify both sides of the inequality.

4x+26>14

Subtract 26 on both sides.

4x+26-26>14-26

Simplify.

4x>-12

Divide by four on both sides. There is no need to switch the sign.

$\frac{4x}{4}>\frac{-12}{4}$

The answer is: x>-3

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Example Question #107 : Inequalities

Solve the inequality: 9(-3x+2)>8(2-3x)

Possible Answers:

$x>\frac{2}{3}$

$x<-\frac{34}{3}$

$x<\frac{34}{3}$

$x<\frac{2}{3}$

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

Correct answer:

$x<\frac{2}{3}$

Explanation:

Use the distributive property to simplify both the left and right sides of the inequality.

9(-3x+2)>8(2-3x)

-27x+18>16-24x

Add 27x on both sides.

-27x+18+27x>16-24x+27x

18>16+3x

Subtract 16 from both sides.

18-16>16+3x-16

2>3x

Divide by three on both sides.

$\frac{2}{3}>\frac{3x}{3}$

$\frac{2}{3}>x$

The answer is: $x<\frac{2}{3}$

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Example Question #108 : Inequalities

Solve: 2x-3>7x-9

Possible Answers:

$x<\frac{6}{5}$

$x<-\frac{4}{3}$

$x<\frac{5}{6}$

$x>\frac{6}{5}$

$x<\frac{4}{3}$

Correct answer:

$x<\frac{6}{5}$

Explanation:

Subtract from both sides.

2x-3-2x>7x-9-2x

-3>5x-9

Add 9 on both sides.

-3+9>5x-9+9

Simplify both sides.

6>5x

Divide both sides by five.

$\frac{6}{5}>\frac{5x}{5}$

$\frac{6}{5}>x$

The answer is: $x<\frac{6}{5}$

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Example Question #109 : Inequalities

Solve: 8(9x-4)>-56

Possible Answers:

$x>\frac{4}{3}$

x>3

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

$x<-\frac{1}{3}$

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

Correct answer:

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

Explanation:

Divide by 8 on both sides.

$\frac{8(9x-4)}{8}>\frac{-56}{8}$

Simplify both sides.

9x-4>-7

Add four on both sides.

9x-4+4>-7+4

9x>-3

Divide by nine on both sides.

$\frac{9x}{9}>\frac{-3}{9}$

The answer is: $x>-\frac{1}{3}$

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Example Question #110 : Inequalities

Simplify the inequality: 9x+8>10x-6

Possible Answers:

$x<-\frac{14}{19}$

x<14

x>14

$x<\frac{14}{19}$

x<-14

Correct answer:

x<14

Explanation:

Isolate the x-variables on one side of the inequality and the integers on the other.

To avoid dividing a negative coefficient and changing the sign, we can subtract on both sides.

9x+8-9x>10x-6-9x

Simplify the inequality.

8>x-6

Add 6 on both sides.

8+6>x-6+6

14>x

The answer is: x<14

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Algebra II : Solving Inequalities

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #101 : Inequalities

Find the solution space for the inequality below:

Example Question #102 : Inequalities

Example Question #103 : Inequalities

Example Question #104 : Inequalities

Example Question #105 : Inequalities

Example Question #106 : Inequalities

Example Question #107 : Inequalities

Example Question #108 : Inequalities

Example Question #109 : Inequalities

Example Question #110 : Inequalities

All Algebra II Resources

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