Hi everyone! Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Lesson 1: The Absolute Value

Topic. This lesson covers Session 1: The Absolute Value.

Learning Outcomes.

  • Solve Absolute Value equations
  • Understand and use interval and inequality notation
  • Solve Absolute Value Inequalities

WeBWorK. There are two WeBWorK assignments on today’s material:

  1. Interval Notation
  2. Absolute Value Inequalities

Additional Video Resources.

Question of the Day: If |x|=5 then what does x equal?

Background Regarding Numbers

VIDEO: Types of Numbers

Video by Irania Vazquez.

Absolute Value

Definition. The absolute value of a real number c, denoted by |c|, is the non-negative number which is equal in magnitude (or size) to c, i.e., is the number resulting from disregarding the sign:

|c|=\left\{ \begin{array}{cl}c, & \text { if } c \text { is positive or zero } \\-c, & \text { if } c \text { is negative }\end{array}

Example 1.

  1. |-4|=,
  2. |4|=
  3. For which real numbers x do you have |x|=3?
  4. Solve for x:|3 x-4|=5
  5. Solve for x:\quad -2 \cdot|12+3 x|=-18

VIDEO: Example 1 – Absolute Value Equations

Video by Irania Vazquez.

Inequalities and Interval Notation

In this section we need to discuss two ideas:

First, the notion of ordering or inequality. For example:

4<9 reads as 4 is less than 9
-3 \leq 2 reads as -3 is less than or equal to 2
\frac{7}{6}>1 reads as \frac{7}{6} is greater than 1
2 \geq-3 reads as 2 is greater than or equal to -3

The second is the called intervalsan interval is a set of numbers on the number line lying (for example) between two endpoints. We can give an interval in three ways: using inequalities, using interval notation, or by drawing a number line. It’s important to be able to describe the same interval in different ways. The following table shows how these three methods are connected:

Table showing inequality notation, number line, and interval notation for various intervals.

Absolute Value Inequalities

To solve an inequality containing an absolute value, we use the following steps:

Step 1: Solve the corresponding equality. The solution of the equality divides the real number line into several subintervals.

Step 2: Using step 1, check the inequality for a number in each of the subintervals. This check determines the intervals of the solution set.
Step 3: Check the endpoints of the intervals.

Example 2. Solve for x :

  • |3 x-5| \geq 11
  • |12-5 x| \leq 1
  • |x+7|\leq -3, |x+7|\geq -3

VIDEO: Example 2 – Absolute Value Inequalities and Interval Notation

Video by Irania Vazquez.

Exit Question

Solve for x: \quad|-3-2x| \geq 7

Answer

In interval notation, the solution set is
S=(-\infty,-5]\cup [2,\infty)

Good job! You are now ready to practice on your own – give the WeBWorK assignment a try. If you get stuck, try using the “Ask for Help” button to ask a question on the Q&A site.