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Algebra 1 : Systems of Inequalities

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Example Questions

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

Give the solution set of the inequality:

-4x - 11 < 57

Possible Answers:

(-17,17)

$(-\infty,-17 )$

The set of all real numbers

$(-17,\infty )$

$(17,\infty )$

Correct answer:

$(-17,\infty )$

Explanation:

-4x - 11 < 57

-4x - 11 + 11 < 57 + 11

-4x < 68

$-4x \div (-4) > 68 \div (-4)$

Note change in direction of the inequality symbol when the expressions are divided by a negative number.

x > -17

or, in interval form,

$(-17,\infty )$

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Example Question #2 : How To Find The Solution To An Inequality With Division

Give the solution set of the inequality:

-4x + 11 < 67

Possible Answers:

$(-\infty ,-14 )$

The set of all real numbers

$(-\infty ,14 )$

$(-14, \infty )$

(-14,14)

Correct answer:

$(-14, \infty )$

Explanation:

-4x + 11 < 67

-4x + 11 - 11 < 67- 11

-4x < 56

$-4x \div (-4) > 56 \div (-4)$

Note change in direction of the inequality symbol when the expressions are divided by a negative number.

x > -14

or, in interval form,

$(-14, \infty )$

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

Solve for :

2-6x>10-2x

Possible Answers:

x<2

x<-2

x>2

x>-2

None of the other answers

Correct answer:

x<-2

Explanation:

First, add and subtract from both sides of the inequality to get -4x>8 .

Then, divide both sides by and reverse the sign since you are dividing by a negative number.

This gives you x<-2 .

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Example Question #2 : How To Find The Solution To An Inequality With Division

Find the solution set to the following compound inequality statement:

$4y + 15 < 91 \textrm{ and } 7y -12 > 86$

Possible Answers:

(14,19)

$(-\infty , 19)$

$(-\infty , \infty )$

$(-\infty , 14) \cup (19, \infty )$

$(14, \infty )$

Correct answer:

(14,19)

Explanation:

Solve each of these two inequalities separately:

4y + 15 < 91

4y + 15 - 15 < 91- 15

4y < 76

$4y \div 4 < 76 \div 4$

y < 19 , or, in interval form, $(-\infty , 19)$

7y -12 > 86

7y -12 + 12 > 86 + 12

7y > 98

$7y \div 7 > 98 \div 7$

y > 14 , or, in interval form, $(14, \infty )$

The two inequalities are connected with an "and", so we take the intersection of the two intervals.

$(14, \infty ) \cap (-\infty , 19) = (14,19)$

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Example Question #9 : How To Find The Solution To An Inequality With Division

Solve for :

$0.4 x - 1.7 \geq -8.5$

Possible Answers:

$[-17, \infty )$

$[1.7, \infty )$

The inequality has no solution.

$[-1.7, \infty )$

$[17, \infty )$

Correct answer:

$[-17, \infty )$

Explanation:

$0.4 x - 1.7 \geq -8.5$

$0.4 x - 1.7 + 1.7 \geq -8.5+ 1.7$

$0.4 x \geq -6.8$

$0.4 x \div 0.4 \geq -6.8\div 0.4$

$x \geq -17$

or, in interval form, $[-17, \infty )$

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Example Question #93 : Equations / Inequalities

Find the solution set of the compound inequality:

$0.3x + 5 > 17 \textrm{ or } 0.7x - 3.1 < 1.6$

Possible Answers:

$(40,\infty )$

(7,40 )

$(-\infty ,40 )$

$(7,\infty )$

$(-\infty ,7) \cup (40,\infty )$

Correct answer:

$(-\infty ,7) \cup (40,\infty )$

Explanation:

Solve each inequality separately:

0.3x + 5 > 17

0.3x + 5 - 5 > 17- 5

0.3x > 12

$0.3x \div 0.3 > 12 \div 0.3$

x > 40

or, in interval form, $(40,\infty )$

0.7x - 3.1 < 1.6

0.7x - 3.1 + 3.1 < 1.6+ 3.1

0.7x < 4.9

$0.7x \div 0.7 < 4.9 \div 0.7$

x < 7

or, in interval form, $(-\infty ,7)$

Since these statements are connected by an "or", we are looking for the union of the intervals. Since the intervals are disjoint, we can simply write this as $(-\infty ,7) \cup (40,\infty )$

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Example Question #91 : Systems Of Inequalities

Find the solution set for :

$5.3 \geq -5x > -8.4$

Possible Answers:

(-1.68, 1.06]

[-1.68, 1.06)

(-1.06, 1.68]

[-1.68, -1.06)

[-1.06, 1.68)

Correct answer:

[-1.06, 1.68)

Explanation:

$5.3 \geq -5x > -8.4$

$5.3 \div (-5 ) \leq -5x \div (-5 ) < -8.4 \div (-5 )$

Note the switch in inequality symbols when the numbers are divided by a negative number.

$-1.06 \leq x < 1.68$

or, in interval form:

[-1.06, 1.68)

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Example Question #13 : How To Find The Solution To An Inequality With Division

Solve for :

7x-2>2x+8

Possible Answers:

x<4

x>2

x<2

x<3

x>3

Correct answer:

x>2

Explanation:

To solve the inequality, you must first separate the integers and the 's. Subtract and add to both sides of the inequality to get

5x>10 .

Then, divide both sides by to get

x>10 .

Since you are not dividing by a negative number, the sign does not need to be reversed.

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Example Question #92 : Systems Of Inequalities

Solve for :

6x-2>4(x+1)

Possible Answers:

x<3

x>3

x>2

None of the other answers

x>3

Correct answer:

x>3

Explanation:

First, use the distributive property to simplify the right side of the inequality:

6x-2>4x+4 .

Then, add and subtract from both sides of the inequality to get

2x>6 .

Finally, divide from both sides to get x>3 .

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Example Question #93 : Systems Of Inequalities

Find the solution set for the inequality.

-2x + 100 < 234

Possible Answers:

Does not exist.

$\left \{ x \left | x > -67\right \}$

$\mathbb{R}$

$\left \{ x | x < - 67\right \}$

$\left \{ x | x < - 67\; or\; x>67\right \}$

Correct answer:

$\left \{ x \left | x > -67\right \}$

Explanation:

-2x + 100 < 234

Subtract 100 from each side:

-2x + 100 - 100 < 234 - 100

-2x < 134

Divide both sides by -2:

x > - 67

(Note that the inequality symbol switched when we divided by a negative number.)

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Algebra 1 : Systems of Inequalities

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

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 #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 #9 : How To Find The Solution To An Inequality With Division

Example Question #93 : Equations / Inequalities

Example Question #91 : Systems Of Inequalities

Example Question #13 : How To Find The Solution To An Inequality With Division

Example Question #92 : Systems Of Inequalities

Example Question #93 : Systems Of Inequalities

All Algebra 1 Resources

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