How to Shade Absolute Value Inequalities?

To shade absolute value inequalities, you essentially follow these steps: graph the boundary line(s) of the absolute value function, then choose a test point to determine which region to shade. Here's a detailed breakdown:

1. Isolate the Absolute Value: If necessary, manipulate the inequality to isolate the absolute value expression on one side. For example, transform 2|x - 1| + 3 < 7 into |x - 1| < 2.

2. Convert to Compound Inequality: Rewrite the absolute value inequality as a compound inequality.

   • For |expression| < value, write -value < expression < value. For example, |x - 1| < 2 becomes -2 < x - 1 < 2.
   • For |expression| > value, write expression < -value OR expression > value. For example, |x - 1| > 2 becomes x - 1 < -2 OR x - 1 > 2.
   • Remember to include "or equal to" signs in your compound inequalities if the original inequality used "≤" or "≥".

3. Solve the Compound Inequality: Solve each inequality in the compound inequality for the variable. In our examples:

   • -2 < x - 1 < 2 becomes -1 < x < 3.
   • x - 1 < -2 OR x - 1 > 2 becomes x < -1 OR x > 3.

4. Graph the Boundary Lines: Graph the solutions on a number line (for one-variable inequalities) or a coordinate plane (for two-variable inequalities where the other variable is often y).

   • One-variable Inequalities (Number Line): Plot the critical values you found in the previous step. Use an open circle for < or > and a closed circle for ≤ or ≥. Then shade the region(s) that satisfy the inequality.
   • Two-variable Inequalities (Coordinate Plane):
     • Replace the inequality sign with an equal sign and graph the resulting equation. This creates the boundary line(s).
     • If the original inequality used < or >, draw a dashed line (or curves). This indicates that points on the line are not included in the solution.
     • If the original inequality used ≤ or ≥, draw a solid line (or curves). This indicates that points on the line are included in the solution.

5. Choose a Test Point: Select a test point that is not on any of the boundary lines. For inequalities in the xy-plane, (0,0) is often a good choice if it's not on the line.

6. Test the Point: Substitute the coordinates of the test point into the original absolute value inequality.

7. Shade the Correct Region:

   • If the test point satisfies the original inequality, shade the region containing the test point.
   • If the test point does not satisfy the original inequality, shade the region that does not contain the test point.

Example:

Consider the inequality y < |x|.

1. The absolute value is already isolated.
2. To graph, consider the equations y = |x|. This is a V-shaped graph with the vertex at (0,0).
3. Because the inequality is <, the boundary lines should be dashed.
4. Choose a test point, like (0, 1).
5. Substitute into the original inequality: 1 < |0| which simplifies to 1 < 0. This is false.
6. Therefore, shade the region below the V-shaped graph, because (0, 1) is above the graph.