How do I solve algebraic inequalities with graphs?

Do you want to make sense of inequalities but find the algebra confusing?

Graphing inequalities turns the math into a visual tool, making it easy to see every solution at a glance.

With a few tricks, you'll soon be shading and sketching your way through every inequality!

Definition: An inequality is a mathematical expression that shows a range of possible values, rather than a single solution (e.g., x > 3).

Graphing inequalities helps visually represent all possible solutions and can be useful in fields like economics, science, and everyday decision-making.

A point is within the solution of the inequality if it lies in the shaded zone.

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Basic Symbols

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Open vs Closed Circles on Number Lines

• Inequalities like ≥ or ≤ use an open circle (the boundary points are included).

• Inequalities with > or < use a closed circle (the boundary points are not included).

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Solid vs Dashed Lines on a Coordinate Plane

• Inequalities like ≥ or ≤ use a solid line (the boundary points are included).

• Inequalities with > or < use a dashed line (the boundary points are not included).

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Image courtesy of Cuemath

1. Draw the number line.

2. Locate and mark the boundary point(s) (c).

3. Use an open circle for > or < and a closed circle for ≥ or ≤.

4. Shade the direction that represents all solutions.

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Example 1: x ≥ −2

x is greater than or equal to -2, so we mark the point -2, shade above -2, and use a closed circle.

Example 2: x < 5

x is less than 5, so we mark the point 5, shade below 5, and use an open circle.

Your turn!

Which number line below represents -3 < x ≤ 1?

A.

B.

C.

D.

1. Graph the boundary line of the inequality as if it were an equation.

2. Use a dashed line for > or < and a solid line for ≥ or ≤.

3. Choose a test point (often the origin) to determine which side of the line to shade (or use the image below).

4. Shade the half-plane that includes all solutions to the inequality.

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Example 1: y < -2

y is less than -2, so we draw the dashed line y = -2 and shade below the line.

Example 2: x ≤ 4

x is less than or equal to 4, so we draw the solid line x = 4 and shade below the line (to the left).

Your turn!

Which graph below represents the inequality y ≥ -3x + 1?

A.

B.

C.

D.

Systems of inequalities: Two or more inequalities graphed on the same coordinate plane, where the solution is the overlapping shaded region.

How to show Ssystems of inequalities visually:

1. Graph each inequality in the system as a separate line with shading.

2. Look for the intersection of all shaded regions, which represents the solution set.

Example: y ≥ x - 1 and y < -x + 3

1. Sketch the solid line y = x - 1 (blue) and the dashed line y = -x + 3 (red) on the same plane.

2. y is greater than or equal to x - 1 so we shade above the blue line.

3. y is less than -x + 3 so we shade below the red line.

4. The solution is the region that is shaded by both inequalities (green).

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Your turn!

Which graph below represents the solution of the inequalities y > 2x and y < x + 1?

A.

B.

C.

D.

Now that you've learned how to graph inequalities, practice applying these skills by graphing a variety of examples!