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Algebra II : Solving Inequalities

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Example Question #81 : Solving Inequalities

Solve the inequality: $3(-2x-7)\leq 8(-6x+2)$

Possible Answers:

$x\leq\frac{ 37}{42}$

$x\leq-\frac{ 5}{42}$

$x\leq\frac{19}{42}$

$x\leq\frac{ 5}{54}$

$x\leq\frac{ 5}{42}$

Correct answer:

$x\leq\frac{ 37}{42}$

Explanation:

Use distribution to simplify both sides.

$3(-2x)-(3)(7)\leq 8(-6x)+8(2)$

$-6x-21\leq -48x+16$

Add 48x on both sides.

$-6x-21+48x\leq -48x+16+48x$

Combine like terms.

$42x-21\leq 16$

Add 21 on both sides of the equation.

$42x-21+21\leq 16+21$

$42x\leq 37$

Divide by 42 on both sides.

$\frac{42x}{42}\leq\frac{ 37}{42}$

The answer is: $x\leq\frac{ 37}{42}$

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Example Question #82 : Solving Inequalities

Solve the inequalities: -3x+2<-41+6x

Possible Answers:

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

$x<\frac{13}{3}$

$x>\frac{43}{9}$

$x<\frac{43}{9}$

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

Correct answer:

$x>\frac{43}{9}$

Explanation:

Subtract from both sides.

-3x+2-6x<-41+6x-6x

Simplify both sides.

-9x+2<-41

Subtract two from both sides of the inequality.

-9x+2-2<-41-2

The inequality becomes:

-9x<-43

Divide by negative nine on both sides. Since we are dividing a negative value, we will need to switch the sign.

$\frac{-9x}{-9}>\frac{-43}{-9}$

The answer is: $x>\frac{43}{9}$

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Example Question #83 : Solving Inequalities

Solve the inequality: 5(x-19)>37

Possible Answers:

$x>\frac{132}{5}$

$x>-\frac{58}{5}$

$x>\frac{204}{5}$

$\textup{The answer is not given.}$

$x>\frac{56}{5}$

Correct answer:

$x>\frac{132}{5}$

Explanation:

Distribute the five with the terms of the binomial.

5(x)-(5)(19)>37

Simplify both terms in parentheses.

5x-95>37

Add 95 on both sides of the inequality.

5x-95+95>37+95

5x>132

Divide by five on both sides.

$\frac{5x}{5}>\frac{132}{5}$

The answer is: $x>\frac{132}{5}$

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Example Question #84 : Solving Inequalities

Solve the inequality: 6-9x>7+10x

Possible Answers:

$x<-\frac{1}{19}$

$x<-\frac{13}{19}$

$x<\frac{13}{19}$

$x>-\frac{13}{19}$

$x>-\frac{1}{19}$

Correct answer:

$x<-\frac{1}{19}$

Explanation:

Add on both sides.

6-9x+9x>7+10x+9x

6>7+19x

Subtract seven from both sides.

6-7>7+19x-7

-1>19x

Divide by 19 on both sides.

$\frac{-1}{19}>\frac{19x}{19}$

Simplify both sides.

The answer is: $x<-\frac{1}{19}$

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Example Question #85 : Solving Inequalities

Solve the inequality: $-3x+4\leq 6-x$

Possible Answers:

$x\leq-5$

$x\geq-5$

$x\geq-1$

$x\leq-\frac{5}{2}$

$x\leq-1$

Correct answer:

$x\geq-1$

Explanation:

Add on both sides.

$-3x+4+3x\leq 6-x+3x$

$4\leq 6+2x$

Subtract six from both sides.

$4-6\leq 6+2x-6$

Simplify both sides.

$-2\leq2x$

Divide by two on both sides. We do not need to change the sign since we are dividing by a positive coefficient.

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

$-1\leq x$

The answer is: $x\geq-1$

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Example Question #86 : Solving Inequalities

Solve the inequality: -3(9-x)>7x-4

Possible Answers:

$x<-\frac{31}{4}$

$x>-\frac{23}{4}$

$x<-\frac{31}{10}$

$x>-\frac{31}{4}$

$x<-\frac{23}{4}$

Correct answer:

$x<-\frac{23}{4}$

Explanation:

Simplify the left side by distribution.

-27+3x>7x-4

Subtract from both sides.

-27+3x-3x>7x-4-3x

-27>4x-4

Add 4 on both sides.

-27+4>4x-4+4

-23>4x

Divide by four on both sides.

$\frac{-23}{4}>\frac{4x}{4}$

The answer is: $x<-\frac{23}{4}$

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Example Question #87 : Solving Inequalities

Solve the inequalities: $\frac{1}{10}x-\frac{1}{100}>\frac{3}{1000}$

Possible Answers:

$x>-\frac{17}{100}$

$x<\frac{7}{100}$

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

$x>\frac{13}{100}$

$x>\frac{103}{100}$

Correct answer:

$x>\frac{13}{100}$

Explanation:

Multiply both sides by the least common denominator in order to eliminate the fractions.

$1000(\frac{1}{10}x-\frac{1}{100})>\frac{3}{1000}\cdot 1000$

The inequality becomes:

100x-10>3

Add on both sides of the inequality.

100x-10+10>3+10

The inequality becomes:

100x>13

Divide by 100 on both sides.

$\frac{100x}{100}>\frac{13}{100}$

The answer is: $x>\frac{13}{100}$

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Example Question #88 : Solving Inequalities

Solve the inequality: 9(-3x+8)>9x+4

Possible Answers:

$x>\frac{19}{9}$

$x>\frac{9}{17}$

$x<\frac{17}{9}$

$x>-\frac{19}{9}$

$\textup{The answer is not given.}$

Correct answer:

$x<\frac{17}{9}$

Explanation:

Expand the terms on the left by distribution.

9(-3x+8) = -27x+72

Rewrite the inequality.

-27x+72>9x+4

Add 27x on both sides.

-27x+72+27x>9x+4+27x

72>36x+4

Subtract 4 from both sides.

72-4>36x+4-4

68>36x

Divide by 36 on both sides.

$\frac{68}{36}>\frac{36x}{36}$

Reduce the fractions.

The answer is: $x<\frac{17}{9}$

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Example Question #89 : Solving Inequalities

Solve the inequality: 9>5x-17

Possible Answers:

$x>-\frac{8}{5}$

$x>\frac{26}{5}$

$x<-\frac{8}{5}$

$x>\frac{8}{5}$

$x<\frac{26}{5}$

Correct answer:

$x<\frac{26}{5}$

Explanation:

Add 17 on both sides.

9+17>5x-17+17

Simplify the inequality.

26>5x

Divide by five on boths ides

$\frac{26}{5}>\frac{5x}{5}$

The answer is: $x<\frac{26}{5}$

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Example Question #90 : Solving Inequalities

Solve the inequality: -9x+5>8(3+x)

Possible Answers:

$x< -\frac{19}{17}$

$x< -\frac{29}{17}$

x>19

$x> -\frac{29}{17}$

$x> -\frac{19}{17}$

Correct answer:

$x< -\frac{19}{17}$

Explanation:

Use the distributive property to expand the right side.

8(3+x) = 8(3)+8x = 8x+24

The inequality becomes: -9x+5>8x+24

To avoid having to divide by a negative value later in the calculation, we should add nine on both sides to move to the right.

-9x+5+9x>8x+24+9x

5>17x+24

Subtract 24 from both sides.

5-24>17x+24-24

-19>17x

Divide by 17 on both sides.

$\frac{-19}{17}>\frac{17x}{17}$

The answer is: $x< -\frac{19}{17}$

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Algebra II : Solving Inequalities

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #81 : Solving Inequalities

Example Question #82 : Solving Inequalities

Example Question #83 : Solving Inequalities

Example Question #84 : Solving Inequalities

Example Question #85 : Solving Inequalities

Example Question #86 : Solving Inequalities

Example Question #87 : Solving Inequalities

Example Question #88 : Solving Inequalities

Example Question #89 : Solving Inequalities

Example Question #90 : Solving Inequalities

All Algebra II Resources

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