How do I solve algebraic inequalities with graphs?

Bart Simpson saying, 'Ooh algebra!' as he studies in his room.

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!

What is an 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.

Graphs depicting y=x, y>x, and y>=x with a point on the lines.

Graphs depicting y=x, y>x, and y>=x with a point above the lines.

Graphs depicting y=x, y>x, and y>=x with a point below the lines. Created by author using Google Slides

Did you know?

The concept of mathematical inequalities dates back to the Greek mathematician Euclid. In The Elements, written around 300 BCE, Euclid presented many ideas about geometric relationships that hinted at inequality concepts. His work indirectly set the stage for understanding inequalities by establishing relationships between figures, a foundational element for modern inequalities.

Understanding Inequality Symbols

Basic Symbols

x<c, x>c, x<=c, and x>=c. Created by author using Google Slides

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).

Number lines depicting closed circles for <= or >= and open circles for < or >. Created by author using Google Slides

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).

Graphs depicting solid lines for <= or >=.

Graphs depicting dashed lines for < or >.

Created by author using Google Slides and Desmos

Graphing One-Variable Inequalities on a Number Line

A number line depicting negative and positive numbers. Image courtesy of Cuemath

Draw the number line.
Locate and mark the boundary point(s) (c).
Use an open circle for > or < and a closed circle for ≥ or ≤.
Shade the direction that represents all solutions.

Shade right of the line for x>c or x>=c. Shade left of the line for x<c or x<=c.

Created by author using Google Slides

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.

Number line with a closed circle at -2. To the right of -2 is shaded.

Example 2: x < 5

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

Number line with an open circle at 5. To the left of 5 is shaded.

Your turn!

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

A. Number line with closed circles at -3 and 1. Between -3 and 1 is shaded.

B. Number line with an open circle at -3 and closed circle at 1. Between -3 and 1 is shaded.

C. Number line with a closed circle at -3 and open circle at 1. Between -3 and 1 is shaded.

D. Number line with open circles at -3 and 1. Between -3 and 1 is shaded.

Quiz

Which number line represents -3 < x ≤ 1?

Graphing on a Coordinate Plane

Graph the boundary line of the inequality as if it were an equation.
Use a dashed line for > or < and a solid line for ≥ or ≤.
Choose a test point (often the origin) to determine which side of the line to shade (or use the image below).
Shade the half-plane that includes all solutions to the inequality.

Shade below the line for y< or y<=. Shade left of the line for x< or x<=. Shade above the line for y> or y>=. Shade right of the line for x> or x>=. Created by author using Google Slides and Desmos

Example 1: y < -2

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

A graph depicting a dashed line at y = -2.

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).

A graph depicting a solid line at x=4.

Your turn!

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

A. A graph depicting a dashed line at y=-3x+1. The left of the line (below the line) is shaded.

B. A graph depicting a dashed line at y=-3x+1. The right of the line (above the line) is shaded.

C. A solid line depicting y=-3x+1. The right of the line (above the line) is shaded.

D. A solid line depicting y=-3x+1. The left of the line (below the line) is shaded.

Quiz

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

Graphing Systems of Inequalities

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:

Graph each inequality in the system as a separate line with shading.
Look for the intersection of all shaded regions, which represents the solution set.

Example: y ≥ x - 1 and y < -x + 3

Sketch the solid line y = x - 1 (blue) and the dashed line y = -x + 3 (red) on the same plane.
y is greater than or equal to x - 1 so we shade above the blue line.
y is less than -x + 3 so we shade below the red line.
The solution is the region that is shaded by both inequalities (green).

A graph depicting y>=x-1 in blue, y<-x+3 in red, and the intersection shaded in green. Created by author using Desmos