Absolute Value Functions: Fresh Take

  • Graph an absolute value function.
  • Solve an absolute value equation.
  • Solve an absolute value inequality.

Understanding Absolute Value

The Main Idea

Absolute value measures distance from zero on the number line. Think of it like this: if you’re standing at zero and you walk 5 steps in either direction, you’ve traveled 5 steps. That’s why both [latex]|5| = 5[/latex] and [latex]|-5| = 5[/latex].

Since distance is never negative, absolute value is never negative. This one idea unlocks everything about absolute value functions, equations, and inequalities.

Mathematically, we can define absolute value as:

[latex]|x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}[/latex]

You can view the transcript for “What is Absolute Value? | Absolute Value Examples | Math with Mr. J” here (opens in new window).

 

Graphing Absolute Value Functions

The Main Idea

The parent absolute value function [latex]f(x) = |x|[/latex] creates a distinctive V-shape with its vertex (the pointy corner) at the origin [latex](0, 0)[/latex]. All absolute value functions have this V-shape, though the vertex can move and the V can flip, stretch, or compress.

The basic graph opens upward with the vertex at [latex](0, 0)[/latex]. Each side of the V has a slope of 1 or -1, creating a perfect symmetric V.

Recall: Transformations work the same way for absolute value functions as they do for other functions. The general form [latex]f(x) = a|x - h| + k[/latex] tells you:

  • The vertex is at [latex](h, k)[/latex]
  • If [latex]a > 0[/latex], the V opens upward
  • If [latex]a < 0[/latex], the V opens downward (it’s flipped)
  • The value of [latex]|a|[/latex] tells you how steep the sides are

You can view the transcript for “Graphing Absolute Value Functions (y=a|x-h|+k)” here (opens in new window).

 

Graph [latex]g(x) = 2|x - 3| + 1[/latex]

The vertex is your starting point. Find [latex](h, k)[/latex] first, plot that point, then determine which direction the V opens and how steep it is.

Solving Absolute Value Equations

The Main Idea

When you see [latex]|A| = B[/latex], you’re asking: “What has a distance of B from zero?” There are usually two answers: the positive and negative versions. That’s why [latex]|x| = 5[/latex] has two solutions: [latex]x = 5[/latex] and [latex]x = -5[/latex].

However, if [latex]B < 0[/latex] (if B is negative), there’s no solution because distance can’t be negative!

Solving [latex]|A| = B[/latex]

  1. Check if [latex]B < 0[/latex]. If yes, there’s no solution.
  2. Isolate the absolute value expression on one side of the equation.
  3. Write two separate equations: [latex]A = B[/latex] and [latex]A = -B[/latex].
  4. Solve each equation.
  5. Check both solutions in the original equation.

You can view the transcript for “How To Solve Absolute Value Equations, Basic Introduction, Algebra” here (opens in new window).

 

Solve [latex]|4x - 1| + 3 = 10[/latex]

Solve [latex]|2x + 5| = -3[/latex]

If the equation simplifies to [latex]|A| = 0[/latex], there’s only ONE solution. Set [latex]A = 0[/latex] and solve. Zero is the only number that’s exactly zero units from zero!

Solving Absolute Value Inequalities

The Main Idea

Absolute value inequalities ask about ranges of distance rather than exact distances. The key is understanding what “less than” versus “greater than” means for distance:

  • [latex]|x| < 5[/latex] means “distance from zero is less than 5” → x is between -5 and 5
  • [latex]|x| > 5[/latex] means “distance from zero is greater than 5” → x is less than -5 OR greater than 5

Think of it this way: “less than” keeps you close (between two values), while “greater than” pushes you far away (outside the range).

You can view the transcript for “How To Solve Absolute Value Inequalities, Basic Introduction, Algebra” here (opens in new window).

 

Solving [latex]|A| < B[/latex] or [latex]|A| \leq B[/latex]

  1. Isolate the absolute value expression.
  2. Rewrite as a compound inequality: [latex]-B < A < B[/latex] (or [latex]-B \leq A \leq B[/latex]).
  3. Solve the compound inequality for the variable.
  4. Write your answer in interval notation.

Solving [latex]|A| > B[/latex] or [latex]|A| \geq B[/latex]

  1. Isolate the absolute value expression.
  2. Write two separate inequalities: [latex]A < -B[/latex] OR [latex]A > B[/latex] (adjust inequality symbols if needed).
  3. Solve each inequality separately.
  4. Write your answer as the union of two intervals using [latex]\cup[/latex].

Solve [latex]|x + 2| \leq 6[/latex]

Solve [latex]|3x - 1| > 8[/latex]

 

The union symbol [latex]\cup[/latex] combines two separate solution sets. We use it when solutions come in two pieces that don’t connect.

Visualizing with Graphs

The Main Idea

You can verify solutions to absolute value inequalities by graphing. Convert your inequality to a two-variable form and graph it—the x-intercepts show the boundary points of your solution, and the shaded region shows which x-values work.

For example, to graph [latex]|x - 2| < 5[/latex]:

  1. Rewrite as [latex]y > |x - 2| - 5[/latex] (move everything to one side)
  2. Graph the inequality using a graphing calculator
  3. The x-intercepts are at [latex]x = -3[/latex] and [latex]x = 7[/latex]
  4. The shaded region confirms your solution [latex](-3, 7)[/latex]

You can view the transcript for “Desmos: Graphing Absolute Value Inequalities and their Solutions” here (opens in new window).