Graphing inequalities on the SAT combines two skills: graphing lines and understanding inequality symbols. Master both and you'll handle these questions confidently.

Step-by-Step Method

  1. Rewrite the inequality in slope-intercept form ( or similar)
  2. Graph the boundary line ()
    • Solid line for or (includes the line)
    • Dashed line for or (excludes the line)
  3. Shade the correct region
    • Shade above for or
    • Shade below for or

Worked Examples

Example 1: Graph

  1. Boundary line: (slope 2, y-intercept )
  2. Dashed line (strict inequality, )
  3. Shade above the line

Any point in the shaded region satisfies the inequality. For instance, : . True! So is in the solution region.

Example 2: Graph

  1. Boundary line: slope , y-intercept
  2. Solid line ( includes the boundary)
  3. Shade below the line

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Converting from Standard Form

Example 3: Graph

First, solve for :


Now graph with a dashed line through with slope , and shade above.

The Test-a-Point Method

If you're ever unsure which side to shade, pick a test point (usually if it's not on the line):

  1. Plug it into the inequality
  2. If the inequality is true, shade the side containing the point
  3. If false, shade the opposite side

Systems of Linear Inequalities

The SAT sometimes shows a graph with two inequalities and asks which point is in the solution region.

Example 4: Which point satisfies both and ?

(A) (B) (C) (D)

Test :

Answer: (C)

Practice Problems

Problem 1: Which inequality is represented by a dashed line through with slope 3, shaded above?

Solution

(dashed = strict inequality, shaded above = greater than)

Problem 2: Does the point satisfy ?

Solution

. Yes, is true.

Problem 3: Rewrite in slope-intercept form and describe the graph.

Solution


(flip the sign when dividing by negative)
Solid line, slope 2, y-intercept , shaded below.

Key Takeaways

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