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Algebra II : Solving Inequalities

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Example Questions

Example Question #331 : Basic Single Variable Algebra

Solve: $\frac{2}{3}x-\frac{8}{9}<\frac{1}{12}$

Possible Answers:

$x<\frac{35}{24}$

$x>\frac{29}{24}$

$x>\frac{35}{24}$

$x<-\frac{29}{24}$

$x>-\frac{29}{24}$

Correct answer:

$x<\frac{35}{24}$

Explanation:

Multiply both sides by the least common denominator. This will save extra work converting all the fractions.

The LCD is 36 since this value is divisible by all three denominators in the inequality.

$36(\frac{2}{3}x-\frac{8}{9})<\frac{1}{12} \cdot 36$

24x-32<3

Add 32 on both sides.

24x-32+32<3+32

24x<35

Divide by 24 on both sides.

$\frac{24x}{24}<\frac{35}{24}$

The answer is: $x<\frac{35}{24}$

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Example Question #2172 : Algebra Ii

Solve the inequality: $3x> \sqrt x$

Possible Answers:

$x\geq0$

x>0

$\textup{There is no solution.}$

$x\leq\frac{1}{9}$

$x>\frac{1}{9}$

Correct answer:

$x>\frac{1}{9}$

Explanation:

Square both sides of the inequality to eliminate the radical.

$(3x)^2> (\sqrt x)^2$

9x^2>x

Subtract from both sides.

9x^2-x>x-x

9x^2-x>0

Factor out an on the left side.

x(9x-1)>0

The factors possible are:

x>0

$9x-1>0 \rightarrow 9x>1 \rightarrow x>\frac{1}{9}$

The values from zero to $\frac{1}{9}$ will not satisfy since the curve will lie beneath the $\sqrt x$ curve.

This means that only $x>\frac{1}{9}$ is valid.

The answer is: $x>\frac{1}{9}$

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Example Question #332 : Basic Single Variable Algebra

Solve the inequalities: $\frac{1}{8}x-\frac{3}{64}\leq \frac{5}{16}$

Possible Answers:

$x\leq\frac{ 17}{8}$

$x\geq-\frac{ 23}{8}$

$x\leq\frac{ 23}{8}$

$x\leq-\frac{ 23}{8}$

$x\geq-\frac{ 17}{8}$

Correct answer:

$x\leq\frac{ 23}{8}$

Explanation:

Multiply both sides by the least common denominator.

The LCD is 64 because this value is divisible by all of the numbers in the denominator.

$64(\frac{1}{8}x-\frac{3}{64})\leq \frac{5}{16} \cdot 64$

$8x-3\leq 20$

Add 3 on both sides.

$8x-3+3\leq 20+3$

$8x\leq 23$

Divide by 8 on both sides.

$\frac{8x}{8}\leq\frac{ 23}{8}$

The answer is: $x\leq\frac{ 23}{8}$

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Example Question #333 : Basic Single Variable Algebra

Solve the inequality: $-\frac{x}{9}> 20$

Possible Answers:

$x>-\frac{20}{9}$

$x<-\frac{20}{9}$

x<-180

x>-180

$x<-\frac{9}{20}$

Correct answer:

x<-180

Explanation:

Multiply by negative nine on both sides of the inequality.

Since we still deal with negative sign, we will need to switch the direction of the sign.

$-\frac{x}{-9}\cdot 9> 20\cdot-9$

The answer is: x<-180

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Example Question #334 : Basic Single Variable Algebra

Solve the inequality: $-3(-2x+3)\leq 14$

Possible Answers:

$x\leq\frac{10}{3}$

$x\leq \frac{23}{6}$

$x<\frac{23}{6}$

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

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

Correct answer:

$x\leq \frac{23}{6}$

Explanation:

Use distribution to simplify the left side.

$-3(-2x)+-3(3)\leq 14$

$6x-9\leq 14$

Add 9 on both sides.

$6x-9+9\leq 14+9$

$6x\leq 23$

Divide by 6 on both sides.

$\frac{6x}{6}\leq \frac{23}{6}$

The answer is: $x\leq \frac{23}{6}$

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Example Question #335 : Basic Single Variable Algebra

Solve the inequality: $-\frac{1}{3}x < 2x-1$

Possible Answers:

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

$x>-\frac{3}{5}$

$x<-\frac{3}{7}$

$x>-\frac{3}{7}$

$x<\frac{3}{7}$

Correct answer:

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

Explanation:

Add $\frac{1}{3}x$ on both sides of the inequality.

$-\frac{1}{3}x +\frac{1}{3}x< 2x-1+\frac{1}{3}x$

Simplify both sides.

$0<\frac{7}{3}x-1$

Add 1 on both sides.

$0+1<\frac{7}{3}x-1+1$

The inequality becomes:

$1<\frac{7}{3}x$

Multiply the reciprocal of the coefficient in front of .

$1\cdot \frac{3}{7}<\frac{7}{3}x \cdot \frac{3}{7}$

$\frac{3}{7}<x$

The answer is: $x>\frac{3}{7}$

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Example Question #341 : Basic Single Variable Algebra

Solve the inequality: 8-3x>16+7x

Possible Answers:

$x<\frac{4}{5}$

x>2

$x<\frac{5}{4}$

$x>\frac{4}{5}$

$x<-\frac{4}{5}$

Correct answer:

$x<-\frac{4}{5}$

Explanation:

Add on both sides.

8-3x+3x>16+7x+3x

The inequality becomes:

8>16+10x

Subtract 16 on both sides.

8-16>16+10x-16

-8>10x

Divide by 10 on both sides.

$\frac{-8}{10}>\frac{10x}{10}$

Reduce both fractions.

$-\frac{4}{5}>x$

The answer is: $x<-\frac{4}{5}$

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Example Question #342 : Basic Single Variable Algebra

Solve: 2(8x+3)>-3-(x+2)

Possible Answers:

$x<-\frac{7}{17}$

$x>-\frac{5}{17}$

$x<-\frac{11}{17}$

$x>\frac{7}{17}$

$x>-\frac{11}{17}$

Correct answer:

$x>-\frac{11}{17}$

Explanation:

Distribute on both sides of the inequality in order to eliminate the parentheses.

2(8x)+2(3)>-3-1(x)-1(2)

The inequality becomes:

16x+6>- 3-x-2

16x+6>-5-x

Add on both sides.

16x+6+x> -5-x+x

17x+6>-5

Subtract 6 from both sides.

17x+6-6>-5-6

17x>-11

Divide by 17 on both sides.

$\frac{17x}{17}>\frac{-11}{17}$

The answer is: $x>-\frac{11}{17}$

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Example Question #343 : Basic Single Variable Algebra

Solve the inequality: $-3x-9\leq 6$

Possible Answers:

$x\leq 1$

$x\leq- 1$

$x\geq- 1$

$x\geq -5$

$x\leq -5$

Correct answer:

$x\geq -5$

Explanation:

Add 9 on both sides.

$-3x-9+9\leq 6+9$

$-3x\leq15$

Divide by negative three on both sides. Dividing by a negative number will switch the sign.

$\frac{-3x}{-3}\leq\frac{15}{-3}$

The answer is: $x\geq -5$

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Example Question #344 : Basic Single Variable Algebra

Solve the inequality: $\frac{2}{3}x+\frac{1}{4}\leq \frac{1}{6}$

Possible Answers:

$x\leq- 8$

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

$x\leq -\frac{1}{8}$

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

$x\leq \frac{1}{8}$

Correct answer:

$x\leq -\frac{1}{8}$

Explanation:

Multiply both sides by the least common denominator to eliminate the fractions. The LCD is 12.

$12(\frac{2}{3}x+\frac{1}{4})\leq \frac{1}{6} \cdot 12$

$8x+3\leq 2$

Subtract three on both sides.

$8x+3-3\leq 2-3$

$8x\leq -1$

Divide both sides by 8.

$\frac{8x}{8}\leq \frac{-1}{8}$

The answer is: $x\leq -\frac{1}{8}$

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Algebra II : Solving Inequalities

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #331 : Basic Single Variable Algebra

Example Question #2172 : Algebra Ii

Example Question #332 : Basic Single Variable Algebra

Example Question #333 : Basic Single Variable Algebra

Example Question #334 : Basic Single Variable Algebra

Example Question #335 : Basic Single Variable Algebra

Example Question #341 : Basic Single Variable Algebra

Example Question #342 : Basic Single Variable Algebra

Example Question #343 : Basic Single Variable Algebra

Example Question #344 : Basic Single Variable Algebra

All Algebra II Resources

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