SAT Math Lesson Plan 5: Absolute Value Equations and Inequalities

Lesson Summary

This 45-minute lesson is part of Media4Math’s 35-lesson SAT Math prep series. It covers absolute value equations, inequalities, and graphing, which are occasionally tested on the SAT—typically 0 to 1 questions per test. Still, understanding absolute value is foundational to algebraic reasoning and supports broader problem-solving skills.

Students will explore how to solve absolute value equations and inequalities both algebraically and graphically. They will also learn how to analyze and sketch the graph of an absolute value function in standard form. The lesson includes detailed examples, a warm-up activity, self-study tips, and a multiple-choice quiz. It supports classroom instruction, tutoring, or independent study.

Lesson Objectives

Understand the definition and properties of absolute value.
Solve absolute value equations algebraically.
Solve absolute value inequalities algebraically and represent solutions graphically.
Graph absolute value functions and interpret their characteristics.

Common Core Standards

CCSS.MATH.CONTENT.HSA.REI.B.3 – Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
CCSS.MATH.CONTENT.HSF.IF.C.7.B – Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Prerequisite Skills

Understanding of linear equations and inequalities
Basic graphing skills on the Cartesian coordinate system
Familiarity with solving simple equations and inequalities

Key Vocabulary

Absolute Value – The distance of a number from zero on the number line, always non-negative. Represented as \( |x| \).
Absolute Value Equation – An equation that contains an absolute value expression, such as \( |x| = a \).
Absolute Value Inequality – An inequality that involves an absolute value expression, such as \( |x| < a \) or \( |x| > a \).
Piecewise Function – A function defined by multiple sub-functions, each applying to a certain interval of the domain. Absolute value functions can be expressed as piecewise functions.

Warm-Up

Goal: Activate prior knowledge of absolute value and linear equations before introducing more complex equations and inequalities.

Part 1: Evaluate Absolute Value Expressions

Evaluate the following:

\( |3| = 3 \)
\( |-5| = 5 \)
\( |0| = 0 \)

What to Notice: Absolute value always returns the non-negative version of a number—it represents distance from zero.

Part 2: Review of Linear Graphs

Before working with absolute value graphs, let’s review how to graph a basic linear function in slope-intercept form: \( y = mx + b \).

Example: Graph the function \( y = 2x - 1 \)

Slope (m) = 2 → Rise over run: up 2, right 1
Y-intercept (b) = -1 → The graph crosses the y-axis at (0, -1)

Plot the y-intercept, then use the slope to find additional points. Draw a straight line through the points extending in both directions.

Use the Desmos graphing calculator to visualize it: Click here to open Desmos

Self-Study Tip: Try graphing different values of m and b to see how the line changes. Pay close attention to how the slope affects the steepness and direction of the line.

Teach

The absolute value of a number represents its distance from zero on the number line. Because distance is always positive (or zero), absolute value expressions like \( |x| \) are always greater than or equal to zero. When solving equations and inequalities that include absolute value, we break them into two cases to account for both the positive and negative possibilities.

Let’s work through five examples that demonstrate how to solve absolute value equations and inequalities algebraically and how to graph them.

Example 1: Solving an Absolute Value Equation

Problem: Solve \( |x - 4| = 7 \)

Step 1: Set up two equations

Note that the expression inside the absolute value sign can be positive or negative:

\( x - 4 = 7 \)
\( x - 4 = -7 \)

Step 2: Solve both

\( x = 11 \)
\( x = -3 \)

Answer: \( x = -3 \) or \( x = 11 \)

Here is the graph of the solution. Note that the points -3 and 11 are the same distance from 4.

Self-Study Tip: Always isolate the absolute value before splitting into two cases.

Example 2: Solving an Absolute Value Inequality (Less Than)

Problem: Solve \( |x + 2| < 5 \)

Step 1: Use the definition for “less than”

\( |A| < B \) means \( -B < A < B \)

Step 2: Set up a compound inequality

\( -5 < x + 2 < 5 \)

Step 3: Solve for x

Subtract 2 from all parts: \( -7 < x < 3 \)

Answer: \( x \in (-7, 3) \)

This graph shows the solution to this inequality.

Self-Study Tip: Inequalities with “less than” often result in an “and” inequality (a segment on the number line).

Example 3: Solving an Absolute Value Inequality (Greater Than)

Problem: Solve \( |2x - 1| \geq 5 \)

Step 1: Use the rule for “greater than”

\( |A| \geq B \) means \( A \leq -B \) or \( A \geq B \)

Step 2: Set up two inequalities

\( 2x - 1 \leq -5 \Rightarrow 2x \leq -4 \Rightarrow x \leq -2 \)
\( 2x - 1 \geq 5 \Rightarrow 2x \geq 6 \Rightarrow x \geq 3 \)

Answer: \( x \leq -2 \) or \( x \geq 3 \)

This graph shows the graph of the inequality.

Self-Study Tip: “Greater than” inequalities lead to two disjoint intervals—watch for that!

Example 4: Graphing an Absolute Value Function

Problem: Graph \( y = |x - 2| \)

Step 1: Identify vertex

Rewrite in the form \( y = |x - h| + k \). Here, \( h = 2 \), \( k = 0 \)
Vertex is at (2, 0)

Step 2: Plot additional points

Try \( x = 1 \): \( y = |1 - 2| = 1 \)
Try \( x = 3 \): \( y = |3 - 2| = 1 \)

Step 3: Sketch

The graph is a V-shape opening upward, vertex at (2, 0)

Answer: Graph has vertex at (2, 0) and is symmetric about the vertical line \( x = 2 \)

Self-Study Tip: Use Desmos to experiment with the transformation of \( y = |x| \) into different forms.

Example 5: Real-World Application

Problem: A machine part must be manufactured with a tolerance of 0.02 inches from a target length of 5 inches. What is the acceptable range for the actual length?

Step 1: Translate into an inequality

\( |x - 5| \leq 0.02 \)

Step 2: Solve the inequality

\( -0.02 \leq x - 5 \leq 0.02 \)

Step 3: Add 5 to all parts

\( 4.98 \leq x \leq 5.02 \)

Answer: The length must be between 4.98 and 5.02 inches

Self-Study Tip: Absolute value is often used in quality control and tolerances—look for key phrases like “within” or “tolerance.”

Example 6: Graphing a Reflected Absolute Value Function

Problem: Graph the function \( y = -|x - 1| + 3 \)

Step 1: Identify the transformation

This is of the form \( y = a|x - h| + k \)
Here, \( a = -1 \), \( h = 1 \), and \( k = 3 \)
The negative sign reflects the graph over the x-axis

Step 2: Find the vertex

The vertex is at \( (1, 3) \)

Step 3: Choose additional points

Let \( x = 0 \): \( y = -|0 - 1| + 3 = -1 + 3 = 2 \)
Let \( x = 2 \): \( y = -|2 - 1| + 3 = -1 + 3 = 2 \)

Step 4: Sketch the graph

Plot the vertex at (1, 3)
Plot (0, 2) and (2, 2) to show the symmetry
The graph opens downward like an upside-down V

Answer: A reflected absolute value graph with vertex at (1, 3) and a maximum value at the vertex.

Self-Study Tip: Anytime you see a negative in front of the absolute value, expect a reflection over the x-axis. Try graphing it in Desmos to verify.

Review

Key Takeaways

The absolute value of a number is its distance from zero and is always non-negative.
Absolute value equations split into two cases: one positive and one negative.
Absolute value inequalities with “less than” form compound inequalities (“and” statements).
Absolute value inequalities with “greater than” form disjoint solutions (“or” statements).
Graphing absolute value functions involves identifying the vertex and plotting a V-shaped graph.

Example 1: Solve an Absolute Value Equation

Problem: Solve \( |3x + 2| = 8 \)

Step 1: Set up two equations

\( 3x + 2 = 8 \Rightarrow 3x = 6 \Rightarrow x = 2 \)
\( 3x + 2 = -8 \Rightarrow 3x = -10 \Rightarrow x = -\frac{10}{3} \)

Answer: \( x = 2 \) or \( x = -\frac{10}{3} \)

Example 2: Solve an Absolute Value Inequality

Problem: Solve \( |x - 4| > 3 \)

Step 1: Use the “greater than” rule

\( |A| > B \Rightarrow A < -B \) or \( A > B \)

Step 2: Apply to the inequality

\( x - 4 < -3 \Rightarrow x < 1 \)
\( x - 4 > 3 \Rightarrow x > 7 \)

Answer: \( x < 1 \) or \( x > 7 \)

Example 3: Graphing a Transformed Absolute Value Function

Problem: Graph \( y = |x + 3| - 2 \)

Step 1: Identify transformation

This is of the form \( y = |x - h| + k \)
Here, \( h = -3 \), \( k = -2 \)

Step 2: Find the vertex

Vertex is at (-3, -2)

Step 3: Plot a few points

\( x = -2 \Rightarrow y = |-2 + 3| - 2 = 1 - 2 = -1 \)
\( x = -4 \Rightarrow y = |-4 + 3| - 2 = 1 - 2 = -1 \)

Answer: V-shaped graph with vertex at (-3, -2), symmetric about the line \( x = -3 \)

Multimedia Resources

To explore the concepts in this lesson further, visit the Multimedia Resources for Lesson 5 page. You’ll find videos, tutorials, and practice activities aligned with this lesson topic.

Quiz

Solve: \( |x - 3| = 5 \)
A) \( x = -2 \) or \( x = 8 \)
B) \( x = -5 \) or \( x = 2 \)
C) \( x = -8 \) or \( x = 2 \)
D) \( x = -3 \) or \( x = 5 \)
Solve: \( |2x + 1| < 7 \)
A) \( x < 3 \)
B) \( x > -4 \)
C) \( -4 < x < 3 \)
D) \( x > 4 \) or \( x < -1 \)
Solve: \( |x| > 4 \)
A) \( x > 4 \)
B) \( x < 4 \)
C) \( x < -4 \) or \( x > 4 \)
D) \( -4 < x < 4 \)
What is the vertex of the function \( y = |x - 2| + 3 \)?
A) (2, 3)
B) (-2, 3)
C) (2, -3)
D) (-2, -3)
Which graph represents a solution to \( |x - 5| \leq 2 \)?
A) Open interval from 3 to 7
B) Closed interval from 3 to 7
C) Open circle at x = 5
D) Rays pointing away from x = 5
Which inequality has no solution?
A) \( |x + 2| < -1 \)
B) \( |x| > 0 \)
C) \( |x - 1| \geq 3 \)
D) \( |x + 5| \leq 8 \)
If \( |x| = a \), where \( a > 0 \), what are the two solutions?
A) \( x = a \)
B) \( x = -a \)
C) \( x = a \) or \( x = -a \)
D) \( x = 0 \)
Solve: \( |x + 4| \geq 6 \)
A) \( x \leq 2 \) or \( x \geq -10 \)
B) \( x \leq -10 \) or \( x \geq 2 \)
C) \( x \leq -2 \) or \( x \geq 10 \)
D) \( x \geq -2 \) and \( x \leq 10 \)
The graph of \( y = |x| \) is:
A) A horizontal line
B) A V-shaped graph with vertex at (0, 0)
C) A U-shaped graph
D) A line through the origin
Solve: \( |3x - 6| = 0 \)
A) \( x = 0 \)
B) \( x = -2 \)
C) \( x = 2 \)
D) \( x = 3 \)

Answer Key

\( x - 3 = 5 \Rightarrow x = 8 \), \( x - 3 = -5 \Rightarrow x = -2 \)
Answer: A
\( |2x + 1| < 7 \Rightarrow -7 < 2x + 1 < 7 \Rightarrow -8 < 2x < 6 \Rightarrow -4 < x < 3 \)
Answer: C
\( |x| > 4 \Rightarrow x < -4 \) or \( x > 4 \)
Answer: C
Vertex is at (2, 3)
Answer: A
\( |x - 5| \leq 2 \Rightarrow 3 \leq x \leq 7 \): closed interval
Answer: B
No absolute value expression is less than a negative number
Answer: A
Two values: \( x = a \) or \( x = -a \)
Answer: C
\( x + 4 \geq 6 \Rightarrow x \geq 2 \), \( x + 4 \leq -6 \Rightarrow x \leq -10 \)
Answer: B
The parent function \( y = |x| \) is V-shaped with vertex at (0, 0)
Answer: B
\( 3x - 6 = 0 \Rightarrow x = 2 \)
Answer: C