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For example, consider the inequality 3 < 7 .

For 3 < 7 , if 8 is added to both sides, we get

3 + 8 < 7 + 8. 11 < 15 True

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For 3 < 7 , if 8 is subtracted from both sides, we get

3 8 < 7 8. 5 < 1 True

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For 3 < 7 , if both sides are multiplied by 8 (a positive number), we get

8 ( 3 ) < 8 ( 7 ) 24 < 56 True

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For 3 < 7 , if both sides are multiplied by 8 (a negative number), we get

( 8 ) 3 > ( 8 ) 7

Notice the change in direction of the inequality sign.

24 > 56 True

If we had forgotten to reverse the direction of the inequality sign we would have obtained the incorrect statement 24 < 56 .

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For 3 < 7 , if both sides are divided by 8 (a positive number), we get

3 8 < 7 8 True

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For 3 < 7 , if both sides are divided by 8 (a negative number), we get

3 8 > 7 8 True ( since .375 .875 )

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Sample set b

Solve the following linear inequalities. Draw a number line and place a point at each solution.

3 x > 15 Divide both sides by 3 . The 3 is a positive number , so we need not reverse the sense of the inequality . x > 5
Thus, all numbers strictly greater than 5 are solutions to the inequality 3 x > 15 .
A number line showing all numbers strictly greater than five.

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2 y 1 16 Add 1 to both sides . 2 y 17 Divide both sides by 2. y 17 2
A number line showing all numbers less than or equal to seventeen over two.

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8 x + 5 < 14 Subtract 5 from both sides . 8 x < 9 Divide both sides by 8. We must reverse the sense of the inequality since we are dividing by a negative number . x > 9 8
A number line showing all numbers strictly greater than negative nine over eight.

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5 3 ( y + 2 ) < 6 y 10 5 3 y 6 < 6 y 10 3 y 1 < 6 y 10 9 y < 9 y > 1
A number line showing all numbers strictly greater than one.

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2 z + 7 4 6 Multiply by 4 2 z + 7 24 Notice the change in the sense of the inequality . 2 z 17 z 17 2
A number line showing all numbers less than or equal to seventeen over two.

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Practice set b

Solve the following linear inequalities.

4 x 1 15

x 4

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5 y + 16 7

y 9 5

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7 ( 4 s 3 ) < 2 s + 8

s < 29 2

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5 ( 1 4 h ) + 4 < ( 1 h ) 2 + 6

h > 1 18

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18 4 ( 2 x 3 ) 9 x

x 30

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3 b 16 4

b 64 3

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7 z + 10 12 < 1

z < 2 7

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x 2 3 5 6

x 3 2

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Compound inequalities

Compound inequality

Another type of inequality is the compound inequality . A compound inequality is of the form:

a < x < b

There are actually two statements here. The first statement is a < x . The next statement is x < b . When we read this statement we say " a is less than x ," then continue saying "and x is less than b ."

Just by looking at the inequality we can see that the number x is between the numbers a and b . The compound inequality a < x < b indicates "betweenness." Without changing the meaning, the statement a < x can be read x > a . (Surely, if the number a is less than the number x , the number x must be greater than the number a .) Thus, we can read a < x < b as " x is greater than a and at the same time is less than b ." For example:

  1. 4 < x < 9 .
    The letter x is some number strictly between 4 and 9. Hence, x is greater than 4 and, at the same time, less than 9. The numbers 4 and 9 are not included so we use open circles at these points.
    A number line showing all numbers strictly greater than four, and strictly less than nine.
  2. 2 < z < 0 .
    The z stands for some number between 2 and 0. Hence, z is greater than 2 but also less than 0.
    A number line showing all numbers strictly greater than negative two, and strictly less than zero.
  3. 1 < x + 6 < 8 .
    The expression x + 6 represents some number strictly between 1 and 8. Hence, x + 6 represents some number strictly greater than 1, but less than 8.
  4. 1 4 5 x 2 6 7 9 .
    The term 5 x 2 6 represents some number between and including 1 4 and 7 9 . Hence, 5 x 2 6 represents some number greater than or equal to 1 4 to but less than or equal to 7 9 .
    A number line showing all numbers greater than or equal to one over four, and less than or equal to seven over nine.

Consider problem 3 above, 1 < x + 6 < 8 . The statement says that the quantity x + 6 is between 1 and 8. This statement will be true for only certain values of x . For example, if x = 1 , the statement is true since 1 < 1 + 6 < 8 . However, if x = 4.9 , the statement is false since 1 < 4.9 + 6 < 8 is clearly not true. The first of the inequalities is satisfied since 1 is less than 10.9 , but the second inequality is not satisfied since 10.9 is not less than 8.

We would like to know for exactly which values of x the statement 1 < x + 6 < 8 is true. We proceed by using the properties discussed earlier in this section, but now we must apply the rules to all three parts rather than just the two parts in a regular inequality.

Sample set c

Solve 1 < x + 6 < 8 .

1 6 < x + 6 6 < 8 6 Subtract 6 from all three parts . 5 < x < 2

Thus, if x is any number strictly between 5 and 2, the statement 1 < x + 6 < 8 will be true.

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Solve 3 < 2 x 7 5 < 8 .

3 ( 5 ) < 2 x 7 5 ( 5 ) < 8 ( 5 ) Multiply each part by 5. 15 < 2 x 7 < 40 Add 7 to all three parts . 8 < 2 x < 47 Divide all three parts by 2. 4 > x > 47 2 Remember to reverse the direction of the inequality signs . 47 2 < x < 4 It is customary (but not necessary) to write the inequality so that inequality arrows point to the left .

Thus, if x is any number between 47 2 and 4, the original inequality will be satisfied.

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Practice set c

Find the values of x that satisfy the given continued inequality.

4 < x 5 < 12

9 < x < 17

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3 < 7 y + 1 < 18

4 7 < y < 17 7

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0 1 6 x 7

1 x 1 6

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5 2 x + 1 3 10

8 x 29 2

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9 < 4 x + 5 2 < 14

23 4 < x < 33 4

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Does 4 < x < 1 have a solution?

no

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Exercises

For the following problems, solve the inequalities.

y + 19 2

y 17

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5 x 20

x 4

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12 b 5 < 24

b > 10

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8 x 5 > 6

x < 15 4

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21 y 8 < 2

y > 16 21

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7 y + 10 4

y 2

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3 x 15 30

x 15

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2 y + 4 3 2 3

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5 ( 2 x 5 ) 15

x 4

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6 ( 3 x 7 ) 48

x 5

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3 ( x + 3 ) > 27

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4 ( y + 3 ) > 0

y < 3

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7 ( x 77 ) 0

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3 x + 2 2 x 5

x 7

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3 x 12 7 x + 4

x 4

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x 4 > 3 x + 12

x > 8

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5 y 14

y 9

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3 [ 4 + 5 ( x + 1 ) ] < 3

x < 2

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2 [ 6 + 2 ( 3 x 7 ) ] 4

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7 [ 3 4 ( x 1 ) ] 91

x 3

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2 ( 4 x 1 ) < 3 ( 5 x + 8 )

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5 ( 3 x 2 ) > 3 ( x 15 ) + 1

x < 2

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Use a calculator to solve this equation. .0091 x 2.885 x 12.014

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What numbers satisfy the condition: twice a number plus one is greater than negative three?

x > 2

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What numbers satisfy the condition: eight more than three times a number is less than or equal to fourteen?

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One number is five times larger than another number. The difference between these two numbers is less than twenty-four. What are the largest possible values for the two numbers? Is there a smallest possible value for either number?

First number: any number strictly smaller that 6.
Second number: any number strictly smaller than 30.
No smallest possible value for either number.
No largest possible value for either number.

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The area of a rectangle is found by multiplying the length of the rectangle by the width of the rectangle. If the length of a rectangle is 8 feet, what is the largest possible measure for the width if it must be an integer (positive whole number) and the area must be less than 48 square feet?

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Exercises for review

( [link] ) Simplify ( x 2 y 3 z 2 ) 5 .

x 10 y 15 z 10

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( [link] ) Simplify [ ( | 8 | ) ] .

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( [link] ) Find the product. ( 2 x 7 ) ( x + 4 ) .

2 x 2 + x 28

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( [link] ) Twenty-five percent of a number is 12.32 . What is the number?

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( [link] ) The perimeter of a triangle is 40 inches. If the length of each of the two legs is exactly twice the length of the base, how long is each leg?

16 inches

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Questions & Answers

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 the state or result of being oxidized
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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