Algebra II : Inequalities

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #2141 : Algebra Ii

Simplify the inequality:  \displaystyle -2(3x-1)>-3(4-x)

Possible Answers:

\displaystyle x>-\frac{7}{9}

\displaystyle x< -\frac{14}{9}

\displaystyle x< -\frac{10}{9}

\displaystyle x< \frac{14}{9}

\displaystyle x>-\frac{14}{9}

Correct answer:

\displaystyle x< \frac{14}{9}

Explanation:

Distribute the negative numbers through the binomials.

\displaystyle -2(3x)-(-2)(1)>-3(4)-(-3)(x)

Simplify the terms.

\displaystyle -6x+2>-12+3x

Add \displaystyle 6x on both sides.

\displaystyle -6x+2+6x>-12+3x+6x

\displaystyle 2>-12+9x

Add 12 on both sides.

\displaystyle 14>9x

Divide by 9 on both sides.

\displaystyle \frac{14}{9}>\frac{9x}{9}

The answer is:  \displaystyle x< \frac{14}{9}

Example Question #2142 : Algebra Ii

Solve the inequality:  \displaystyle \frac{1}{3}x-3>12

Possible Answers:

\displaystyle x>3

\displaystyle x>45

\displaystyle x< 45

\displaystyle x< 3

\displaystyle x< 4

Correct answer:

\displaystyle x>45

Explanation:

Add three on both sides.

\displaystyle \frac{1}{3}x-3+3>12+3

The inequality becomes:

\displaystyle \frac{1}{3}x>15

Multiply by three on both sides.

\displaystyle \frac{1}{3}x\cdot 3>15\cdot 3

The answer is:  \displaystyle x>45

Example Question #2143 : Algebra Ii

Solve the inequality:  \displaystyle 9x-18>56

Possible Answers:

\displaystyle x>\frac{23}{3}

\displaystyle x>\frac{38}{9}

\displaystyle x< \frac{74}{9}

\displaystyle x>\frac{74}{9}

\displaystyle x< \frac{48}{9}

Correct answer:

\displaystyle x>\frac{74}{9}

Explanation:

Add 18 on both sides.

\displaystyle 9x-18+18>56+18

Simplify both sides.

\displaystyle 9x>74

Divide by nine on both sides.

\displaystyle \frac{9x}{9}>\frac{74}{9}

The answer is:  \displaystyle x>\frac{74}{9}

Example Question #2144 : Algebra Ii

Solve the inequality:  \displaystyle 8x\leq -9x+7

Possible Answers:

\displaystyle x\geq7

\displaystyle x\leq7

\displaystyle x\leq-7

\displaystyle x\leq\frac{7}{17}

\displaystyle x\geq\frac{7}{17}

Correct answer:

\displaystyle x\leq\frac{7}{17}

Explanation:

Add \displaystyle 9x on both sides of the inequality.

\displaystyle 8x+9x\leq -9x+7+9x

The inequality becomes:

\displaystyle 17x\leq7

Divide by seventeen on both sides.

\displaystyle \frac{17x}{17}\leq\frac{7}{17}

The answer is:  \displaystyle x\leq\frac{7}{17}

Example Question #2145 : Algebra Ii

Solve the inequality:  \displaystyle -\frac{1}{4}x+6\leq 10

Possible Answers:

\displaystyle x\leq-64

\displaystyle x\leq-16

\displaystyle x\geq-1

\displaystyle x\leq-1

\displaystyle x\geq-16

Correct answer:

\displaystyle x\geq-16

Explanation:

Subtract six from both sides.

\displaystyle -\frac{1}{4}x+6-6\leq 10-6

\displaystyle -\frac{1}{4}x\leq4

Multiply by negative four on both sides.  Note that since this is similar to divide by a negative value, we will have to change the direction of the sign.

\displaystyle -\frac{1}{4}x \cdot -4\geq4 \cdot-4

The answer is:  \displaystyle x\geq-16

Example Question #2151 : Algebra Ii

Solve the inequality:  \displaystyle -9x-3>-8(x-3)

Possible Answers:

\displaystyle x< -21

\displaystyle x>27

\displaystyle x< -27

\displaystyle x< -33

\displaystyle x>33

Correct answer:

\displaystyle x< -27

Explanation:

Simplify the right side of the inequality.

\displaystyle -9x-3>-8x+24

Add \displaystyle 9x on both sides of the inequality.

\displaystyle -9x-3+9x>-8x+24+9x

\displaystyle -3>x+24

Subtract 24 from both sides.

\displaystyle -3-24>x+24-24

Simplify both sides.

The answer is:  \displaystyle x< -27

Example Question #111 : Inequalities

Solve the inequality:  \displaystyle -10x(-2)>-2+31x

Possible Answers:

\displaystyle x< -\frac{2}{19}

\displaystyle x< \frac{2}{11}

\displaystyle x>-\frac{2}{19}

\displaystyle x< -\frac{2}{11}

\displaystyle x>0

Correct answer:

\displaystyle x< \frac{2}{11}

Explanation:

Be careful not to subtract or assume subtraction of the terms on the left side.  The left side is a product of two values.

\displaystyle 20x>-2+31x

Subtract \displaystyle 31x on both sides.

\displaystyle 20x-31x>-2+31x-31x

The inequality becomes: 

\displaystyle -11x>-2

Divide both sides by negative eleven.  We will have to change the inequality sign.

\displaystyle \frac{-11x}{-11}>\frac{-2}{-11}

The answer is:  \displaystyle x< \frac{2}{11}

Example Question #73 : Solving Inequalities

Solve the following inequality:  \displaystyle -3(-2x+7)< -2(5-6x)

Possible Answers:

\displaystyle x>-\frac{11}{6}

\displaystyle x< \frac{11}{18}

\displaystyle x< \frac{11}{6}

\displaystyle x< \frac{31}{18}

\displaystyle x< \frac{31}{6}

Correct answer:

\displaystyle x>-\frac{11}{6}

Explanation:

Use distribution to simplify both sides.

\displaystyle -3(-2x)+-3(7)< -2(5)-(-2)(6x)

Simplify all the terms in the inequality.

\displaystyle 6x-21< -10+12x

Subtract \displaystyle 6x on both sides.  This will allow us to avoid changing the sign.

\displaystyle 6x-21-6x< -10+12x-6x

\displaystyle -21< 6x-10

Add 10 on both sides.

\displaystyle -21+10< 6x-10+10

\displaystyle -11< 6x

Divide by 6 on both sides.

\displaystyle \frac{-11}{6}< \frac{6x}{6}

\displaystyle \frac{-11}{6}< x

The answer is:  \displaystyle x>-\frac{11}{6}

Example Question #111 : Inequalities

Solve the inequality:  \displaystyle -8x-9>-36

Possible Answers:

\displaystyle x>-\frac{45}{8}

\displaystyle x< -\frac{45}{8}

\displaystyle x< \frac{27}{8}

\displaystyle x< \frac{25}{4}

\displaystyle x>\frac{27}{8}

Correct answer:

\displaystyle x< \frac{27}{8}

Explanation:

Add nine on both sides.

\displaystyle -8x-9+9>-36+9

\displaystyle -8x>-27

Divide by negative eight on both sides.  We will need to switch the sign since we are dividing a negative value.

\displaystyle \frac{-8x}{-8}< \frac{-27}{-8}

The answer is:  \displaystyle x< \frac{27}{8}

Example Question #311 : Basic Single Variable Algebra

Solve the inequality:  \displaystyle -4(-x+2)< 6-4x

Possible Answers:

\displaystyle x< -\frac{1}{4}

\displaystyle x>\frac{7}{4}

\displaystyle x>-\frac{1}{4}

\displaystyle \textup{No solution.}

\displaystyle x< \frac{7}{4}

Correct answer:

\displaystyle x< \frac{7}{4}

Explanation:

Simplify the left side of the inequality by distribution.

\displaystyle -4(-x)+(-4)(2)< 6-4x

\displaystyle 4x-8< 6-4x

Add \displaystyle 4x on both sides.

\displaystyle 4x-8+4x< 6-4x+4x

The inequality becomes:

\displaystyle 8x-8< 6

Add 8 on both sides.

\displaystyle 8x-8+8< 6+8

\displaystyle 8x< 14

Divide by eight on both sides.

\displaystyle \frac{8x}{8}< \frac{14}{8}

Reduce the fractions.

The answer is:  \displaystyle x< \frac{7}{4}

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