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Algebra 1 : How to find the solution to an inequality with division

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Example Question #11 : How To Find The Solution To An Inequality With Division

Find the solution set of the compound inequality:

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

Possible Answers:

(7,40 )

$(40,\infty )$

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

$(7,\infty )$

$(-\infty ,40 )$

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

Find the solution set for :

$5.3 \geq -5x > -8.4$

Possible Answers:

[-1.06, 1.68)

(-1.06, 1.68]

[-1.68, 1.06)

(-1.68, 1.06]

[-1.68, -1.06)

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

Solve for :

6x-2>4(x+1)

Possible Answers:

x<3

x>3

x>2

x>3

None of the other answers

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

Find the solution set for the inequality.

-2x + 100 < 234

Possible Answers:

$\mathbb{R}$

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

Does not exist.

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

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

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

Solve for :

3x-1< x+5

Possible Answers:

x>1

x<1

None of the other answers are correct.

x>3

x<3

Correct answer:

x<3

Explanation:

To solve this inequality, get the 's to one side of the equation and the integers to the other side.

Add 1 to both sides:

2x<6

Divide both sides by 2:

x<3

(Since we are not dividing by a negative number, there is no need to reverse the sign.)

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Example Question #14 : How To Find The Solution To An Inequality With Division

Find the solution set to the compound inequality:

$\left ( 5y+19< 74 \right )\bigcap \left ( -3y+13> 40 \right )$

Possible Answers:

$(-\infty ,11)$

$\left ( -\infty , -9\right )$

(-9,11)

$(-\infty ,-9) \cup (11, \infty )$

$\left ( -\infty , \infty \right )$

Correct answer:

$\left ( -\infty , -9\right )$

Explanation:

Solve each of these two inequalities separately:

5y + 19 < 74

5y + 19 -19 < 74 -19

5y < 55

$5y \div 5 < 55 \div 5$

y < 11 , or in interval form, $\left ( -\infty , 11\right )$

-3y + 13 > 40

-3y + 13 - 13 > 40- 13

-3y > 27

$-3y \div (-3) < 27 \div (-3)$

(Note the flipping of the inequality because of the division by a negative number.)

y < -9 , or in interval form, $\left ( -\infty , -9\right )$

The question asks about the intersection of the two intervals:

$\left ( -\infty , 11\right ) \cap \left ( -\infty , -9\right )$

The intersection is the area of the number line that the two sets share, or $\left ( -\infty , -9\right )$ .

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Example Question #18 : How To Find The Solution To An Inequality With Division

Solve the following inequality.

5x+4 > x-4

Possible Answers:

x<3

x>-2

x>4

x>5

x>2

Correct answer:

x>-2

Explanation:

To solve this inequality, move all the terms with on one side and all other terms on the other side, then solve for x.

5x+4 > x-4

4x>-8

x>-2

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Example Question #19 : How To Find The Solution To An Inequality With Division

Solve the inequality

2x+9> -x+1 .

Possible Answers:

$x>-\frac{8}{3}\:$

$x<\frac{8}{3}\:$

$x>-\frac{3}{8}\:$

$x>\frac{8}{3}\:$

$x<-\frac{8}{3}\:$

Correct answer:

$x>-\frac{8}{3}\:$

Explanation:

Inequalities are treated as equalities when it comes to balancing, with the exception of division by a negative number (then flip the greater/less than symbol).

The question is asking for the solved inequality isolating . Combine like terms across sides by adding or subtracting same value to both sides.

2x+9> -x+1

2x>-x-8

3x>-8

$x>-\frac{8}{3}$

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Example Question #20 : How To Find The Solution To An Inequality With Division

Solve the inequality.

$4x+3\geq -4$

Possible Answers:

$x\leq -\frac{7}{4}$

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

$x> -\frac{7}{4}$

$x\geq -\frac{7}{4}$

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

Correct answer:

$x\geq -\frac{7}{4}$

Explanation:

Use the properties of inequalities to balance the inequality and isolate .

$4x+3\geq -4$

First subtract three from both sides.

$4x\geq -7$

Next, divide by four.

$x\geq -\frac{7}{4}$

Since no division or multiplication of a negative number occurred, the inequality sign remains the same.

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Algebra 1 : How to find the solution to an inequality with division

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #11 : How To Find The Solution To An Inequality With Division

Example Question #11 : How To Find The Solution To An Inequality With Division

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

Example Question #11 : How To Find The Solution To An Inequality With Division

Example Question #12 : How To Find The Solution To An Inequality With Division

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

Example Question #14 : How To Find The Solution To An Inequality With Division

Example Question #18 : How To Find The Solution To An Inequality With Division

Example Question #19 : How To Find The Solution To An Inequality With Division

Example Question #20 : How To Find The Solution To An Inequality With Division

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