Linear Inequalities and Absolute Value Inequalities

Learning Objectives

In this section, you will:

• Use interval notation
• Use properties of inequalities.
• Solve inequalities in one variable algebraically.
• Solve absolute value inequalities.

Figure 1

It is not easy to make the honor roll at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.

Using Interval Notation

Indicating the solution to an inequality such as x≥4𝑥≥4 can be achieved in several ways.

We can use a number line as shown in Figure 2. The blue ray begins at x=4𝑥=4 and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.

Figure 2

We can use set-builder notation: {x|x≥4},{𝑥|𝑥≥4}, which translates to “all real numbers x such that x is greater than or equal to 4.” Notice that braces are used to indicate a set.

The third method is interval notation, in which solution sets are indicated with parentheses or brackets. The solutions to x≥4𝑥≥4 are represented as [4,∞).[4,∞). This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.

The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval, or a set of numbers in which a solution falls, are [−2,6),[−2,6), or all numbers between −2−2 and 6,6, including −2,−2, but not including 6;6; (−1,0),(−1,0), all real numbers between, but not including −1−1 and 0;0; and (−∞,1],(−∞,1], all real numbers less than and including 1.1. Table 1 outlines the possibilities.

Set Indicated	Set-Builder Notation	Interval Notation
All real numbers between a and b, but not including a or b	{x|a<x<b}{𝑥|𝑎<𝑥<𝑏}	(a,b)(𝑎,𝑏)
All real numbers greater than a, but not including a	{x|x>a}{𝑥|𝑥>𝑎}	(a,∞)(𝑎,∞)
All real numbers less than b, but not including b	{x|x<b}{𝑥|𝑥<𝑏}	(−∞,b)(−∞,𝑏)
All real numbers greater than a, including a	{x|x≥a}{𝑥|𝑥≥𝑎}	[a,∞)[𝑎,∞)
All real numbers less than b, including b	{x|x≤b}{𝑥|𝑥≤𝑏}	(−∞,b](−∞,𝑏]
All real numbers between a and b, including a	{x|a≤x<b}{𝑥|𝑎≤𝑥<𝑏}	[a,b)[𝑎,𝑏)
All real numbers between a and b, including b	{x|a<x≤b}{𝑥|𝑎<𝑥≤𝑏}	(a,b](𝑎,𝑏]
All real numbers between a and b, including a and b	{x|a≤x≤b}{𝑥|𝑎≤𝑥≤𝑏}	[a,b][𝑎,𝑏]
All real numbers less than a or greater than b	{x|x<aorx>b}{𝑥|𝑥<𝑎or𝑥>𝑏}	(−∞,a)∪(b,∞)(−∞,𝑎)∪(𝑏,∞)
All real numbers	{x|xis all real numbers}{𝑥|𝑥is all real numbers}	(−∞,∞)(−∞,∞)

EXAMPLE 1

Using Interval Notation to Express All Real Numbers Greater Than or Equal to a

Use interval notation to indicate all real numbers greater than or equal to −2.−2.

TRY IT #1

Use interval notation to indicate all real numbers between and including −3−3 and 5.5.

EXAMPLE 2

Using Interval Notation to Express All Real Numbers Less Than or Equal to a or Greater Than or Equal to b

Write the interval expressing all real numbers less than or equal to −1−1 or greater than or equal to 1.1

TRY IT #2

Express all real numbers less than −2−2 or greater than or equal to 3 in interval notation.

Using the Properties of Inequalities

When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.

PROPERTIES OF INEQUALITIES

AdditionPropertyMultiplicationPropertyIfa<b,thena+c<b+c.Ifa<bandc>0,thenac<bc.Ifa<bandc<0,thenac>bc.𝐴𝑑𝑑𝑖𝑡𝑖𝑜𝑛𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦If𝑎<𝑏,then𝑎+𝑐<𝑏+𝑐.𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦If𝑎<𝑏and𝑐>0,then𝑎𝑐<𝑏𝑐.If𝑎<𝑏and𝑐<0,then𝑎𝑐>𝑏𝑐.

These properties also apply to a≤b,𝑎≤𝑏, a>b,𝑎>𝑏, and a≥b.𝑎≥𝑏.

EXAMPLE 3

Demonstrating the Addition Property

Illustrate the addition property for inequalities by solving each of the following:

1. ⓐ x−15<4𝑥−15<4
2. ⓑ 6≥x−16≥𝑥−1
3. ⓒ x+7>9𝑥+7>9

TRY IT #3

Solve: 3x−2<1.3𝑥−2<1.

EXAMPLE 4

Demonstrating the Multiplication Property

Illustrate the multiplication property for inequalities by solving each of the following:

1. ⓐ 3x<63𝑥<6
2. ⓑ −2x−1≥5−2𝑥−1≥5
3. ⓒ 5−x>105−𝑥>10

TRY IT #4

Solve: 4x+7≥2x−3.4𝑥+7≥2𝑥−3.

Solving Inequalities in One Variable Algebraically

As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.

EXAMPLE 5

Solving an Inequality Algebraically

Solve the inequality: 13−7x≥10x−4.13−7𝑥≥10𝑥−4.

TRY IT #5

Solve the inequality and write the answer using interval notation: −x+4<12x+1.−𝑥+4<12𝑥+1.

EXAMPLE 6

Solving an Inequality with Fractions

Solve the following inequality and write the answer in interval notation: −34x≥−58+23x.−34𝑥≥−58+23𝑥.

TRY IT #6

Solve the inequality and write the answer in interval notation: −56x≤34+83x.−56𝑥≤34+83𝑥.

Understanding Compound Inequalities

A compound inequality includes two inequalities in one statement. A statement such as 4<x≤64<𝑥≤6 means 4<x4<𝑥 and x≤6.𝑥≤6. There are two ways to solve compound inequalities: separating them into two separate inequalities or leaving the compound inequality intact and performing operations on all three parts at the same time. We will illustrate both methods.

EXAMPLE 7

Solving a Compound Inequality

Solve the compound inequality: 3≤2x+2<6.3≤2𝑥+2<6.

TRY IT #7

Solve the compound inequality: 4<2x−8≤10.4<2𝑥−8≤10.

EXAMPLE 8

Solving a Compound Inequality with the Variable in All Three Parts

Solve the compound inequality with variables in all three parts: 3+x>7x−2>5x−10.3+𝑥>7𝑥−2>5𝑥−10.

TRY IT #8

Solve the compound inequality: 3y<4−5y<5+3y.3𝑦<4−5𝑦<5+3𝑦.

Solving Absolute Value Inequalities

As we know, the absolute value of a quantity is a positive number or zero. From the origin, a point located at (−x,0)(−𝑥,0) has an absolute value of x,𝑥, as it is x units away. Consider absolute value as the distance from one point to another point. Regardless of direction, positive or negative, the distance between the two points is represented as a positive number or zero.

An absolute value inequality is an equation of the form

|A|<B,|A|≤B,|A|>B,or|A|≥B,|𝐴|<𝐵,|𝐴|≤𝐵,|𝐴|>𝐵,or|𝐴|≥𝐵,

Where A, and sometimes B, represents an algebraic expression dependent on a variable x. Solving the inequality means finding the set of all x𝑥 –values that satisfy the problem. Usually this set will be an interval or the union of two intervals and will include a range of values.

There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two equations. The advantage of the algebraic approach is that solutions are exact, as precise solutions are sometimes difficult to read from a graph.

Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of x-values such that the distance between x𝑥 and 600 is less than or equal to 200. We represent the distance between x𝑥 and 600 as |x−600|,|𝑥−600|, and therefore, |x−600|≤200|𝑥−600|≤200 or

−200≤x−600≤200−200+600≤x−600+600≤200+600400≤x≤800−200≤𝑥−600≤200−200+600≤𝑥−600+600≤200+600400≤𝑥≤800

This means our returns would be between $400 and $800.

To solve absolute value inequalities, just as with absolute value equations, we write two inequalities and then solve them independently.

ABSOLUTE VALUE INEQUALITIES

For an algebraic expression X, and k>0,𝑘>0, an absolute value inequality is an inequality of the form

|X|<kis equivalent to −k<X<k|X|>kis equivalent to X<−kor X>k|𝑋|<𝑘is equivalent to −𝑘<𝑋<𝑘|𝑋|>𝑘is equivalent to 𝑋<−𝑘or 𝑋>𝑘

These statements also apply to |X|≤k|𝑋|≤𝑘 and |X|≥k.|𝑋|≥𝑘.

EXAMPLE 9

Determining a Number within a Prescribed Distance

Describe all values x𝑥 within a distance of 4 from the number 5.

TRY IT #9

Describe all x-values within a distance of 3 from the number 2.

EXAMPLE 10

Solving an Absolute Value Inequality

Solve |x−1|≤3|𝑥−1|≤3 .

EXAMPLE 11

Using a Graphical Approach to Solve Absolute Value Inequalities

Given the equation y=−12∣∣4x−5∣∣+3,𝑦=−12|4𝑥−5|+3, determine the x-values for which the y-values are negative

TRY IT #10

Solve −2|k−4|≤−6.