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Linear inequalities in One Variable

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules ((Reference)) and ((Reference))). Objectives of this module: understand the meaning of inequalities, be able to recognize linear inequalities, know, and be able to work with, the algebra of linear inequalities and with compound inequalities.

Overview

  • Inequalities
  • Linear Inequalities
  • The Algebra of Linear Inequalities
  • Compound Inequalities

Inequalities

Relationships of Inequality

We have discovered that an equation is a mathematical way of expressing the relationship of equality between quantities. Not all relationships need be relationships of equality, however. Certainly the number of human beings on earth is greater than 20. Also, the average American consumes less than 10 grams of vitamin C every day. These types of relationships are not relationships of equality, but rather, relationships of inequality.

Linear Inequalities

Linear Inequality

A linear inequality is a mathematical statement that one linear expression is greater than or less than another linear expression.

Inequality Notation

The following notation is used to express relationships of inequality:
 > Strictly greater than < Strictly less than ≥ Greater than or equal to ≤ Less than or equal to > Strictly greater than < Strictly less than ≥ Greater than or equal to ≤ Less than or equal to

Note that the expression x>12 x>12 has infinitely many solutions. Any number strictly greater than 12 will satisfy the statement. Some solutions are 13, 15, 90, 12.1, 16.3 and 102.51 12.1, 16.3 and 102.51 .

Sample Set A

The following are linear inequalities in one variable.

Example 1

  1. x≤12 x≤12
  2. x+7>4 x+7>4
  3. y+3≥2y−7 y+3≥2y−7
  4. P+26<10(4P−6) P+26<10(4P−6)
  5. 2r−9 5 >15 2r−9 5 >15

The following are not linear inequalities in one variable.

Example 2

  1. x 2 <4 x 2 <4 .
    The term x 2 x 2 is quadratic, not linear.
  2. x≤5y+3 x≤5y+3 .
    There are two variables. This is a linear inequality in two variables.
  3. y+1≠5 y+1≠5 .
    Although the symbol ≠ ≠ certainly expresses an inequality, it is customary to use only the symbols <, >, ≤, ≥ <, >, ≤, ≥ .

Practice Set A

A linear equation, we know, may have exactly one solution, infinitely many solutions, or no solution. Speculate on the number of solutions of a linear inequality. (Hint: Consider the inequalities x<x−6 x<x−6 and x≥9 x≥9 .)

A linear inequality may have infinitely many solutions, or no solutions.

The Algebra of Linear Inequalities

Inequalities can be solved by basically the same methods as linear equations. There is one important exception that we will discuss in item 3 of the algebra of linear inequalities.

The Algebra of Linear Inequalities

Let a, b, and c a, b, and c represent real numbers and assume that
 a<b (or a>b) a<b (or a>b)
Then,  if a<b,  if a<b,

  1. a+c<b+c and a−c<b−c a+c<b+c and a−c<b−c .
    If any real number is added to or subtracted from both sides of an inequality, the sense of the inequality remains unchanged.
  2. If c c is a positive real number, then if a<b, a<b,
    ac<bc and a c < b c . ac<bc and a c < b c .
    If both sides of an inequality are multiplied or divided by the same positive number the sense of the inequality remains unchanged.
  3. If c c is a negative real number, then if a<b, a<b,
    ac>bc and a c > b c . ac>bc and a c > b c .
    If both sides of an inequality are multiplied or divided by the same negative number, the inequality sign must be reversed (change direction) in order for the resulting inequality to be equivalent to the original inequality. (See problem 4 in the next set of examples.)

For example, consider the inequality 3<7 3<7 .

Example 3

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

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

Example 4

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

3−8<7−8. −5<−1 True 3−8<7−8. −5<−1 True

Example 5

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

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

Example 6

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

(−8)3>(−8)7 (−8)3>(−8)7

Notice the change in direction of the inequality sign.

−24>−56 True −24>−56 True

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

Example 7

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

3 8 < 7 8 True 3 8 < 7 8 True

Example 8

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

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

Sample Set B

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

Example 9

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

Example 10

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

Example 11

−8x+5<14 Subtract 5 from both sides. −8x<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 −8x+5<14 Subtract 5 from both sides. −8x<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.

Example 12

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

Example 13

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

Practice Set B

Solve the following linear inequalities.

Exercise 1

y−6≤5 y−6≤5

Solution

y≤11 y≤11

Exercise 2

x+4>9 x+4>9

Solution

x>5 x>5

Exercise 3

4x−1≥15 4x−1≥15

Solution

x≥4 x≥4

Exercise 4

−5y+16≤7 −5y+16≤7

Solution

y≥ 9 5 y≥ 9 5

Exercise 5

7(4s−3)<2s+8 7(4s−3)<2s+8

Solution

s< 29 2 s< 29 2

Exercise 6

5(1−4h)+4<(1−h)2+6 5(1−4h)+4<(1−h)2+6

Solution

h> 1 18 h> 1 18

Exercise 7

18≥4(2x−3)−9x 18≥4(2x−3)−9x

Solution

x≥−30 x≥−30

Exercise 8

− 3b 16 ≤4 − 3b 16 ≤4

Solution

b≥ −64 3 b≥ −64 3

Exercise 9

−7z+10 −12 <−1 −7z+10 −12 <−1

Solution

z<− 2 7 z<− 2 7

Exercise 10

−x− 2 3 ≤ 5 6 −x− 2 3 ≤ 5 6

Solution

x≥ −3 2 x≥ −3 2

Compound Inequalities

Compound Inequality

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

a<x<b a<x<b

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

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

  1. 4<x<9 4<x<9 .
    The letter x x is some number strictly between 4 and 9. Hence, x 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 −2<z<0 .
    The z z stands for some number between −2 −2 and 0. Hence, z z is greater than −2 −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 1<x+6<8 .
    The expression x+6 x+6 represents some number strictly between 1 and 8. Hence, x+6 x+6 represents some number strictly greater than 1, but less than 8.
  4. 1 4 ≤ 5x−2 6 ≤ 7 9 1 4 ≤ 5x−2 6 ≤ 7 9 .
    The term 5x−2 6 5x−2 6 represents some number between and including 1 4 1 4 and 7 9 7 9 . Hence, 5x−2 6 5x−2 6 represents some number greater than or equal to 1 4 1 4 to but less than or equal to 7 9 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 1<x+6<8 . The statement says that the quantity x+6 x+6 is between 1 and 8. This statement will be true for only certain values of x x . For example, if x=1 x=1 , the statement is true since 1<1+6<8 1<1+6<8 . However, if x=4.9 x=4.9 , the statement is false since 1<4.9+6<8 1<4.9+6<8 is clearly not true. The first of the inequalities is satisfied since 1 is less than 10.9 10.9 , but the second inequality is not satisfied since 10.9 10.9 is not less than 8.

We would like to know for exactly which values of x x the statement 1<x+6<8 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

Example 14

Solve 1<x+6<8 1<x+6<8 .

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

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

Example 15

Solve −3< −2x−7 5 <8 −3< −2x−7 5 <8 .

−3(5)< −2x−7 5 (5)<8(5) Multiply each part by 5. −15<−2x−7<40 Add 7 to all three parts. −8<−2x<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. −3(5)< −2x−7 5 (5)<8(5) Multiply each part by 5. −15<−2x−7<40 Add 7 to all three parts. −8<−2x<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 x is any number between − 47 2 − 47 2 and 4, the original inequality will be satisfied.

Practice Set C

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

Exercise 11

4<x−5<12 4<x−5<12

Solution

9<x<17  9<x<17 

Exercise 12

−3<7y+1<18 −3<7y+1<18

Solution

− 4 7 <y< 17 7 − 4 7 <y< 17 7

Exercise 13

0≤1−6x≤7 0≤1−6x≤7

Solution

−1≤x≤ 1 6 −1≤x≤ 1 6

Exercise 14

−5≤ 2x+1 3 ≤10 −5≤ 2x+1 3 ≤10

Solution

−8≤x≤ 29 2 −8≤x≤ 29 2

Exercise 15

9< −4x+5 −2 <14 9< −4x+5 −2 <14

Solution

23 4 <x< 33 4 23 4 <x< 33 4

Exercise 16

Does 4<x<−1  4<x<−1  have a solution?

Solution

no

Exercises

For the following problems, solve the inequalities.

Exercise 17

x+7<12 x+7<12

Solution

x<5 x<5

Exercise 18

y−5≤8 y−5≤8

Exercise 19

y+19≥2 y+19≥2

Solution

y≥−17 y≥−17

Exercise 20

x−5>16 x−5>16

Exercise 21

3x−7≤8 3x−7≤8

Solution

x≤3 x≤3

Exercise 22

9y−12≤6 9y−12≤6

Exercise 23

2z+8<7 2z+8<7

Solution

z<− 1 2 z<− 1 2

Exercise 24

4x−14>21 4x−14>21

Exercise 25

−5x≤20 −5x≤20

Solution

x≥−4 x≥−4

Exercise 26

−8x<40 −8x<40

Exercise 27

−7z<77 −7z<77

Solution

z>−11 z>−11

Exercise 28

−3y>39 −3y>39

Exercise 29

x 4 ≥12 x 4 ≥12

Solution

x≥48 x≥48

Exercise 30

y 7 >3 y 7 >3

Exercise 31

2x 9 ≥4 2x 9 ≥4

Solution

x≥18 x≥18

Exercise 32

5y 2 ≥15 5y 2 ≥15

Exercise 33

10x 3 ≤4 10x 3 ≤4

Solution

x≤ 6 5 x≤ 6 5

Exercise 34

−5y 4 <8 −5y 4 <8

Exercise 35

−12b 5 <24 −12b 5 <24

Solution

b>−10 b>−10

Exercise 36

−6a 7 ≤−24 −6a 7 ≤−24

Exercise 37

8x −5 >6 8x −5 >6

Solution

x<− 15 4 x<− 15 4

Exercise 38

14y −3 ≥−18 14y −3 ≥−18

Exercise 39

21y −8 <−2 21y −8 <−2

Solution

y> 16 21 y> 16 21

Exercise 40

−3x+7≤−5 −3x+7≤−5

Exercise 41

−7y+10≤−4 −7y+10≤−4

Solution

y≥2 y≥2

Exercise 42

6x−11<31 6x−11<31

Exercise 43

3x−15≤30 3x−15≤30

Solution

x≤15 x≤15

Exercise 44

−2y+ 4 3 ≤− 2 3 −2y+ 4 3 ≤− 2 3

Exercise 45

5(2x−5)≥15 5(2x−5)≥15

Solution

x≥4 x≥4

Exercise 46

4(x+1)>−12 4(x+1)>−12

Exercise 47

6(3x−7)≥48 6(3x−7)≥48

Solution

x≥5 x≥5

Exercise 48

3(−x+3)>−27 3(−x+3)>−27

Exercise 49

−4(y+3)>0 −4(y+3)>0

Solution

y<−3 y<−3

Exercise 50

−7(x−77)≤0 −7(x−77)≤0

Exercise 51

2x−1<x+5 2x−1<x+5

Solution

x<6 x<6

Exercise 52

6y+12≤5y−1 6y+12≤5y−1

Exercise 53

3x+2≤2x−5 3x+2≤2x−5

Solution

x≤−7 x≤−7

Exercise 54

4x+5>5x−11 4x+5>5x−11

Exercise 55

3x−12≥7x+4 3x−12≥7x+4

Solution

x≤−4 x≤−4

Exercise 56

−2x−7>5x −2x−7>5x

Exercise 57

−x−4>−3x+12 −x−4>−3x+12

Solution

x>8 x>8

Exercise 58

3−x≥4 3−x≥4

Exercise 59

5−y≤14 5−y≤14

Solution

y≥−9 y≥−9

Exercise 60

2−4x≤−3+x 2−4x≤−3+x

Exercise 61

3[4+5(x+1)]<−3 3[4+5(x+1)]<−3

Solution

x<−2 x<−2

Exercise 62

2[6+2(3x−7)]≥4 2[6+2(3x−7)]≥4

Exercise 63

7[−3−4(x−1)]≤91 7[−3−4(x−1)]≤91

Solution

x≥−3 x≥−3

Exercise 64

−2(4x−1)<3(5x+8) −2(4x−1)<3(5x+8)

Exercise 65

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

Solution

x<−2 x<−2

Exercise 66

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

Exercise 67

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

Solution

x>−2 x>−2

Exercise 68

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

Exercise 69

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?

Solution

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.

Exercise 70

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?

Exercises for Review

Exercise 71

((Reference)) Simplify ( x 2 y 3 z 2 ) 5 ( x 2 y 3 z 2 ) 5 .

Solution

x 10 y 15 z 10 x 10 y 15 z 10

Exercise 72

((Reference)) Simplify −[−(−| −8 |)] −[−(−| −8 |)] .

Exercise 73

((Reference)) Find the product. (2x−7) (x+4) (2x−7) (x+4) .

Solution

2 x 2 +x−28 2 x 2 +x−28

Exercise 74

((Reference)) Twenty-five percent of a number is 12.32 12.32 . What is the number?

Exercise 75

((Reference)) 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?

Solution

16 inches

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