4.4: Absolute Value Inequalities - Mathematics LibreTexts

Solving |x| < a

The solutions of \[|x|<a \nonumber \] again depend upon the value and sign of the number a. To solve |x| < a graphically, we must determine where the graph of the left-hand side lies below the graph of the right-hand side of the inequality |x| < a. There are three cases to consider.

• Case I: a < 0

In this case, the graph of y = a lies strictly below the x-axis. As you can see in Figure \(\PageIndex{1}\)(a), the graph of y = |x| never lies below the graph of y = a. Hence, the inequality |x| < a has no solutions.

• Case II: a = 0

In this case, the graph of y = 0 coincides with the x-axis. As you can see in Figure \(\PageIndex{1}\)(b), the graph of y = |x| never lies strictly below the x-axis. Hence, the inequality |x| < 0 has no solutions.

• Case III: a > 0

In this case, the graph of y = a lies strictly above the x-axis. In Figure \(\PageIndex{1}\)(c), the graph of y = |x| and y = a intersect at x = −a and x = a. In Figure \(\PageIndex{1}\)(c), we also see that the graph of y = |x| lies strictly below the graph of y = a when x is in-between −a and a; that is, when −a < x < a.

In Figure \(\PageIndex{1}\)(c), we’ve dropped dashed vertical lines from the points of intersection of the two graphs to the x-axis. On the x-axis, we’ve shaded the solution of |x| < a, that is, −a < x < a.

This discussion leads to the following key property.

property 1

The solution of |x| < a depends upon the value and sign of a.

• Case I: a < 0

The inequality |x| < a has no solution.

• Case II: a = 0

The inequality |x| < 0 has no solution.

• Case III: a > 0

The inequality |x| < a has solution set {x : −a < x < a}.

Let’s look at some examples.

Example \(\PageIndex{1}\)

Solve the inequality |x| < −5 for x.

Solution

The graph of the left-hand side of |x| < −5 is the “V” of Figure \(\PageIndex{1}\)(a). The graph of the right-hand side of |x| < −5 is a horizontal line located 5 units below the x-axis. This is the situation shown in Figure \(\PageIndex{1}\)(a). The graph of y = |x| is therefore never below the graph of y = −5. Thus, the inequality |x| < −5 has no solution.

An alternate approach is to consider the fact that the absolute value of x is always nonnegative and can never be less than −5. Thus, the inequality |x| < −5 has no solution.

Example \(\PageIndex{2}\)

Solve the inequality |x| < 0 for x.

Solution

This is the case shown in Figure \(\PageIndex{1}\)(b). The graph of y = |x| is never strictly below the x-axis. Thus, the inequality |x| < 0 has no solution.

Example \(\PageIndex{3}\)

Solve the inequality |x| < 8 for x.

Solution

The graph of the left-hand side of |x| < 8 is the “V” of Figure \(\PageIndex{1}\)(c). The graph of the right-hand side of |x| < 8 is a horizontal line located 8 units above the x-axis. This is the situation depicted in Figure \(\PageIndex{1}\)(c). The graphs intersect at (−8, 8) and (8, 8) and the graph of y = |x| lies strictly below the graph of y = 8 for values of x in-between −8 and 8. Thus, the solution of |x| < 8 is −8 < x < 8.

It helps the intuition if you check the results of the last example. Note that numbers between −8 and 8, such as −7.75, −3 and 6.8 satisfy the inequality,

\[|-7.75|<8 \qquad \text { and } \quad|-3|<8 \quad \text { and } \quad|6.8|<8 \nonumber \]

while values that do not lie between −8 and 8 do not satisfy the inequality. For example, none of the numbers −9.3, 8.2, and 11.7 lie between −8 and 8, and each of the following is a false statement.

\[|-9.3|<8 \quad \text { and } \qquad|8.2|<8 \qquad \text { and } \qquad|11.7|<8 \quad \text { (all are false) } \nonumber \]

If you reflect upon these results, they will help cement the notion that the solution of |x| < 8 is all values of x satisfying −8 < x < 8.

Example \(\PageIndex{4}\)

Solve the inequality |5 − 2x| < −3 for x.

Solution

If the inequality were |x| < −3, we would not hesitate. This is the situation depicted in Figure \(\PageIndex{1}\)(a) and the inequality |x| < −3 has no solutions. The reasoning applied to |x| < −3 works equally well for the inequality |5 − 2x| < −3. The left-hand side of this inequality must be nonnegative, so its graph must lie on or above the x-axis. The right-hand side of |5 − 2x| < −3 is a horizontal line located 3 units below the x-axis. Therefore, the graph of y = |5 − 2x| can never lie below the graph of y = −3 and the inequality |5 − 2x| < −3 has no solution.

We can verify this result with the graphing calculator. Load the left- and right-hand sides of |5 − 2x| < −3 into Y1 and Y2, respectively, as shown in Figure \(\PageIndex{2}\)(a). From the ZOOM menu, select 6:ZStandard to produce the image shown in Figure \(\PageIndex{2}\)(b).

As predicted, the graph of y = |5 − 2x| never lies below the graph of y = −3, so the inequality |5 − 2x| < −3 has no solution.

Example \(\PageIndex{5}\)

Solve the inequality |5 − 2x| < 0 for x.

Solution

We know that the left-hand side of the inequality |5 − 2x| < 0 has the “V” shape indicated in Figure \(\PageIndex{1}\)(b). The graph “touches” the x-axis when |5 − 2x| = 0, or when

\[\begin{aligned} 5-2 x &=0 \\-2 x &=-5 \\ x &=\frac{5}{2} \end{aligned} \nonumber \]

However, the graph of y = |5 − 2x| never falls below the x-axis, so the inequality |5 − 2x| < 0 has no solution.

Intuitively, it should be clear that the inequality |5−2x| < 0 has no solution. Indeed, the left-hand side of this inequality is always nonnegative, and can never be strictly less than zero.

Example \(\PageIndex{6}\)

Solve the inequality |5 − 2x| < 3 for x.

Solution

In this example, the graph of the right-hand side of the inequality |5 − 2x| < 3 is a horizontal line located 3 units above the x-axis. The graph of the left-hand side of the inequality has the “V” shape shown in Figure \(\PageIndex{3}\)(b) and (c). You can use the intersect utility on the graphing calculator to find the points of intersection of the graphs of y = |5 − 2x| and y = 3, as we have done in Figures \(\PageIndex{3}\)(b) and (c). Note that the calculator indicates two points of intersection, one at x = 1 and a second at x = 4.

The graph of y = |5 − 2x| falls below the graph of y = 3 for all values of x between 1 and 4. Hence, the solution of the inequality |5 − 2x| < 3 is the set of all x satisfying 1 < x < 4; i.e. {x : 1 < x < 4}.

Expectations:

We need a way of summarizing this graphing calculator approach on our homework paper. First, draw a reasonable facsimile of your calculator’s viewing window on your homework paper. Use a ruler to draw all lines. Complete the following checklist.

• Label each axis, in this case with x and y.
• Scale each axis. To do this, press the WINDOW button on your calculator, then report the values of xmin, xmax, ymin, and ymax on the appropriate axis.
• Label each graph with its equation.
• Drop dashed vertical lines from the points of intersection to the x-axis. Shade and label the solution set of the inequality on the x-axis.

Following the guidelines in the above checklist, we obtain the image in Figure \(\PageIndex{4}\).

Algebraic Approach. Let’s now explore an algebraic solution of the inequality |5 − 2x| < 3. Much as |x| < 3 implies that −3 < x < 3, the inequality

\[|5-2 x|<3 \nonumber \]

requires that

\[-3<5-2 x<3 \nonumber \]

We can subtract 5 from all three members of this last inequality, then simplify.

\[\begin{aligned}-3-5 &<5-2 x-5<3-5 \\ &-8<-2 x<-2 \end{aligned} \nonumber \]

Divide all three members of this last inequality by −2, reversing the inequality symbols as you go.

\[4>x>1 \nonumber \]

We prefer that our inequalities read from “small-to-large,” so we write

\[1<x<4 \nonumber \]

This form matches the order of the shaded solution on the number line in Figure \(\PageIndex{4}\), which we found using the graphing calculator.

The algebraic technique of this last example leads us to the following property.

Property 8

If a > 0, then the inequality |x| < a is equivalent to the inequality −a < x < a.

This property provides a simple method for solving inequalities of the form |x| < a. Let’s apply this algebraic technique in the next example.

Example \(\PageIndex{7}\)

Solve the inequality |4x + 5| < 7 for x.

Solution

The first step is to use Property 8 to write that \[|4 x+5|<7 \nonumber \]

is equivalent to the inequality

\[-7<4 x+5<7 \nonumber \]

From here, we can solve for x by first subtracting 5 from all three members, then dividing through by 4.

\[\begin{array}{l}{-12<4 x<2} \\ {-3<x<\frac{1}{2}}\end{array} \nonumber \]

We can sketch the solution on a number line.

And we can describe the solution in both interval and set-builder notation as follows.

\[\left(-3, \frac{1}{2}\right)=\left\{x :-3<x<\frac{1}{2}\right\} \nonumber \]

Assuming that a > 0, the inequality \(|x| \leq a\) requires that we find where the absolute value of x is either “less than” a or “equal to” a. We know that |x| < a when −a < x < a and we know that |x| = a when x = −a or x = a. Thus, the solution of \(|x| \leq a\) is the “union” of these two solutions.

This argument leads to the following property.

Property 10

If \(a > 0\), then the inequality \(|x| \leq a\) is equivalent to the inequality \(−a \leq x \leq a\).

Example \(\PageIndex{8}\)

Solve the inequality \(5 − 3|x − 4| \geq −4\) for x.

Solution

At first glance, the inequality \[5-3|x-4| \geq-4 \nonumber \] has a form quite dissimilar from what we’ve done thus far. However, let’s subtract 5 from both sides of the inequality.

\[-3|x-4| \geq-9 \nonumber \]

Now, let’s divide both sides of this last inequality by −3, reversing the inequality sign.

\[|x-4| \leq 3 \nonumber \]

Aha! Familiar ground. Using Property 10, this last inequality is equivalent to

\[-3 \leq x-4 \leq 3 \nonumber \]

and when we add 4 to all three members, we have the solution.

\[1 \leq x \leq 7 \nonumber \]

We can sketch the solution on a number line

And we can describe the solution with interval and set-builder notation.

\[[1,7]=\{x : 1 \leq x \leq 7\} \nonumber \]