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Precalculus for Team-Based Inquiry Learning: 2024 Development 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 \(\lvert x \rvert=-x\text{.}\)

Activity 1.4.2.

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

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 \(0\text{.}\)
(a)
What values on the number line could \(x-7 \) equal?
  1. \(\displaystyle x=-7\)
  2. \(\displaystyle x=-2\)
  3. \(\displaystyle x=0\)
  4. \(\displaystyle x=2\)
  5. \(\displaystyle x=7\)
Answer.
B and D
(b)
This gives us two separate equations to solve. What are those two equations?
  1. \(\displaystyle x-7=-7\)
  2. \(\displaystyle x-7=-2\)
  3. \(\displaystyle x-7=0\)
  4. \(\displaystyle x-7=2\)
  5. \(\displaystyle x-7=7\)
Answer.
B and D
(c)
Solve each equation for \(x\text{.}\)
Answer.
\(x=-5\) and \(x=9\)

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*}

Activity 1.4.5.

Solve the following absolute value equations.
(a)
\(\lvert 3x+4 \rvert = 10\)
  1. \(\displaystyle \{-2, 2\}\)
  2. \(\displaystyle \left\{-\frac{14}{3}, 2\right\}\)
  3. \(\displaystyle \{-10, 10\}\)
  4. No solution
Answer.
B
(b)
\(3\lvert x-7 \rvert+5 = 11\)
  1. \(\displaystyle \{-2, 2\}\)
  2. \(\displaystyle \{-9, 9\}\)
  3. \(\displaystyle \{5, 9\}\)
  4. No solution
Answer.
C
(c)
\(2\lvert x+1 \rvert+8 = 4\)
  1. \(\displaystyle \{-4, 4\}\)
  2. \(\displaystyle \{-6, 6\}\)
  3. \(\displaystyle \{5, 7\}\)
  4. No solution
Answer.
D

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.

Activity 1.4.7.

Just as with linear equations and inequalities, we can consider absolute value inequalities from equations.
(a)
Which values are solutions to the absolute value inequality \(\lvert x-7 \rvert \le 2\text{?}\)
  1. \(\displaystyle x=9\)
  2. \(\displaystyle x=7\)
  3. \(\displaystyle x=5\)
  4. \(\displaystyle x=-9\)
Answer.
A, B and C
(b)
Rewrite the absolute value inequality \(\lvert x-7 \rvert \le 2\) as a compound inequality.
  1. \(\displaystyle 0 \le x-7 \le 2\)
  2. \(\displaystyle -2 \le x-7 \le 2\)
  3. \(\displaystyle -2 \le x-7 \le 0\)
  4. \(\displaystyle 2 \le x \le 7\)
Answer.
B
(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. \(\displaystyle [7,9]\)
  2. \(\displaystyle [5,9]\)
  3. \(\displaystyle [5,7]\)
  4. \(\displaystyle [2,7]\)
Answer.
B
(d)
Draw the solution to \(\lvert x-7 \rvert \le 2\) on the number line.
Answer.
B

Activity 1.4.8.

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

Definition 1.4.9.

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

Activity 1.4.10.

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

Subsection 1.4.2 Videos

It would be great to include videos down here, like in the Calculus book!
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