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Precalculus for Team-Based Inquiry Learning 2025 Edition — Instructor Version

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Section 1.4 Absolute Value Equations and Inequalities (EQ4)

Subsection 1.4.1 Activities

Remark 1.4.1.

An absolute value, written \(\lvert x \rvert\text{,}\) is the non-negative value of \(x\text{.}\) If \(x\) is a positive number, then \(\lvert x \rvert=x\text{.}\) If \(x\) is a negative number, then \(\lvert x \rvert=-x\text{.}\)
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Activity 1.4.2.

Let’s consider how to solve an equation when an absolute value is involved.
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(a)
Which values are solutions to the absolute value equation \(\lvert x \rvert = 2\text{?}\)
  1. \(\displaystyle x=2\)
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  2. \(\displaystyle x=0\)
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  3. \(\displaystyle x=-1\)
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  4. \(\displaystyle x=-2\)
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Answer.
A and D
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(b)
Which values are solutions to the absolute value equation \(\lvert x-7 \rvert = 2\text{?}\)
  1. \(\displaystyle x=9\)
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  2. \(\displaystyle x=7\)
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  3. \(\displaystyle x=5\)
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  4. \(\displaystyle x=-9\)
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Answer.
A and C
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(c)
Which values are solutions to the absolute value equation \(3\lvert x-7 \rvert +5= 11\text{?}\) It may be helpful to rewrite the equation to isolate the absolute value.
  1. \(\displaystyle x=7\)
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  2. \(\displaystyle x=-9\)
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  3. \(\displaystyle x=5\)
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  4. \(\displaystyle x=9\)
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Answer.
C and D
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Activity 1.4.3.

Absolute value represents the distance a value is from 0 on the number line. So, \(\lvert x-7 \rvert = 2\) means that the expression \(x-7\) is \(2\) units away from \(0\text{.}\)
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(b)
This gives us two separate equations to solve. What are those two equations?
  1. \(\displaystyle x-7=-7\)
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  2. \(\displaystyle x-7=-2\)
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  3. \(\displaystyle x-7=0\)
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  4. \(\displaystyle x-7=2\)
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  5. \(\displaystyle x-7=7\)
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Answer.
B and D
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(c)
Solve each equation for \(x\text{.}\)
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Answer.
\(x=-5\) and \(x=9\)
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Remark 1.4.4.

When solving an absolute value equation, begin by isolating the absolute value expression. Then rewrite the equation into two linear equations and solve. If \(c \gt 0\text{,}\)
\begin{equation*} \lvert ax+b \rvert = c \end{equation*}
becomes the following two equations
\begin{equation*} ax+b =c \quad \text{and} \quad ax+b=-c \end{equation*}
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Activity 1.4.5.

Solve the following absolute value equations.
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Remark 1.4.6.

Since the absolute value represents a distance, it is always a positive number. Whenever you encounter an isolated absolute value equation equal to a negative value, there will be no solution.
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Activity 1.4.7.

Just as with linear equations and inequalities, we can consider absolute value inequalities from equations.
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(a)
Which values are solutions to the absolute value inequality \(\lvert x-7 \rvert \le 2\text{?}\)
  1. \(\displaystyle x=9\)
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  2. \(\displaystyle x=7\)
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  3. \(\displaystyle x=5\)
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  4. \(\displaystyle x=-9\)
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Answer.
A, B and C
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(b)
Rewrite the absolute value inequality \(\lvert x-7 \rvert \le 2\) as a compound inequality.
  1. \(\displaystyle 0 \le x-7 \le 2\)
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  2. \(\displaystyle -2 \le x-7 \le 2\)
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  3. \(\displaystyle -2 \le x-7 \le 0\)
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  4. \(\displaystyle 2 \le x \le 7\)
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Answer.
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(c)
Solve the compound inequality that is equivalent to \(\lvert x-7 \rvert \le 2\) found in part (b). Write the solution in interval notation.
  1. \(\displaystyle [7,9]\)
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  2. \(\displaystyle [5,9]\)
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  3. \(\displaystyle [5,7]\)
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  4. \(\displaystyle [2,7]\)
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Answer.
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Activity 1.4.8.

Now let’s consider another type of absolute value inequality.
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(a)
Which values are solutions to the absolute value inequality \(\lvert x-7 \rvert \ge 2\text{?}\)
  1. \(\displaystyle x=9\)
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  2. \(\displaystyle x=7\)
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  3. \(\displaystyle x=5\)
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  4. \(\displaystyle x=-9\)
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Answer.
A, C and D
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(b)
Which two of the following inequalities are equivalent to \(\lvert x-7 \rvert \ge 2\text{.}\)
  1. \(\displaystyle x-7 \le 2 \)
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  2. \(\displaystyle x-7 \le -2\)
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  3. \(\displaystyle x-7 \ge 2\)
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  4. \(\displaystyle x-7 \ge -2\)
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Answer.
B and C
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(c)
Solve the two inequalities found in part (b). Write the solution in interval notation and graph on the number line.
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  1. \(\displaystyle (-\infty,7] \cup [9,\infty)\)
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  2. \(\displaystyle (-\infty,5] \cup [9,\infty)\)
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  3. \(\displaystyle (-\infty,5] \cup [7,\infty)\)
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  4. \(\displaystyle (-\infty,2] \cup [7,\infty)\)
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Answer.
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Definition 1.4.9.

When solving an absolute value inequality, rewrite it as compound inequalities. Assume \(k\) is positive. \(\lvert x \rvert \lt k \text{ becomes } -k \lt x \lt k\text{.}\) \(\lvert x \rvert \gt k \text{ becomes } x\gt k \text{ or } x\lt-k\text{.}\)
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Activity 1.4.10.

Solve the following absolute value inequalities. Write your solution in interval notation and graph on a number line.
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(a)
\(\lvert 3x+4 \rvert \lt 10\)
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Answer.
\(-\dfrac{14}{3} \lt x \lt 2\text{,}\) \(\left( -\dfrac{14}{3}, 2 \right)\)
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(b)
\(3\lvert x-7 \rvert+5 \gt 11\)
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Answer.
\(x \lt 5\) and \(x \gt 9\text{,}\) \((-\infty, 5) \cup (9, \infty)\)
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Subsection 1.4.2 Exercises

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