How do you graph an absolute value inequality?

Graphing an absolute value inequality involves finding all the numbers that satisfy the inequality and representing them on a number line. The result will either be a segment between two points or two rays extending in opposite directions from two points.

Here's a step-by-step approach:

Isolate the Absolute Value: If necessary, rewrite the inequality so the absolute value expression is alone on one side. For example, if you have |x + 2| - 3 < 5, first add 3 to both sides to get |x + 2| < 8.
Create Two Inequalities: An absolute value inequality like |x| < a (where 'a' is a positive number) is equivalent to -a < x < a. Similarly, |x| > a is equivalent to x < -a or x > a. Apply this principle to your specific inequality.
- For |x + 2| < 8: This becomes -8 < x + 2 < 8.
- For |x - 1| > 3: This becomes x - 1 < -3 or x - 1 > 3.
Solve Each Inequality: Solve each of the inequalities you created in the previous step.
- For -8 < x + 2 < 8: Subtract 2 from all parts: -10 < x < 6.
- For x - 1 < -3 or x - 1 > 3: Add 1 to both sides of each inequality: x < -2 or x > 4.
Graph the Solution Set:
- Number Line: Draw a number line.
- Endpoints: Locate the endpoint(s) of your solution(s) on the number line.
  - -10 < x < 6: Place open circles at -10 and 6. Open circles indicate that the endpoints are not included in the solution (because the inequalities are strict: < or >).
  - x < -2 or x > 4: Place open circles at -2 and 4.
- Shading: Shade the region(s) of the number line that represent the solutions.
  - -10 < x < 6: Shade the region between -10 and 6. This segment represents all numbers greater than -10 and less than 6.
  - x < -2 or x > 4: Shade the region to the left of -2 and the region to the right of 4. These rays represent all numbers less than -2 or greater than 4.
- Closed Circles: If the original inequality had a "less than or equal to" (≤) or "greater than or equal to" (≥) sign, use closed circles (or brackets) at the endpoints to indicate that those values are included in the solution.

Example Summary Table:

Inequality	Equivalent Inequality(ies)	Solution	Graph on Number Line
`\|x\| < 3`	`-3 < x < 3`	`-3 < x < 3`	(---o========o---) where 'o' represents an open circle at -3 and 3, and '========' represents the shaded region between them.
`\|x\| ≤ 3`	`-3 ≤ x ≤ 3`	`-3 ≤ x ≤ 3`	(---[========]---) where '[' and ']' represent closed brackets at -3 and 3, and '========' represents the shaded region between them.
`\|x\| > 3`	`x < -3` or `x > 3`	`x < -3` or `x > 3`	(====o-------o====) where 'o' represents an open circle at -3 and 3, and '====' represents the shaded regions to the left of -3 and to the right of 3.
`\|x\| ≥ 3`	`x ≤ -3` or `x ≥ 3`	`x ≤ -3` or `x ≥ 3`	(====[-------]====) where '[' and ']' represent closed brackets at -3 and 3, and '====' represents the shaded regions to the left of -3 and to the right of 3.
`\|x + 1\| < 2`	`-2 < x + 1 < 2`	`-3 < x < 1`	(---o========o---) where 'o' represents an open circle at -3 and 1, and '========' represents the shaded region between them.
`\|x - 2\| > 1`	`x - 2 < -1` or `x - 2 > 1`	`x < 1` or `x > 3`	(====o-------o====) where 'o' represents an open circle at 1 and 3, and '====' represents the shaded regions to the left of 1 and to the right of 3.

In conclusion, graphing an absolute value inequality involves isolating the absolute value, creating two related inequalities, solving those inequalities, and then representing the solution set as a segment or two rays on a number line, taking care to use open or closed circles/brackets based on the inequality symbols.