Graphing Linear
Inequalities
Linear Inequalities
• A linear inequality is similar to a
linear equation, but the equal sign is
replaced with an inequality symbol.
• A solution of a linear inequality is
any ordered pair that makes the
inequality true.
Example 1: Identifying
Solutions of Inequalities
Tell whether the ordered pair is a solution of
the inequality.
(–2, 4); y < 2x + 1
y < 2x + 1
4 2(–2) + 1
Substitute (–2, 4) for (x, y).
4 –4 + 1
4 < –3 
(–2, 4) is not a solution.
Example 2: Identifying
Solutions of Inequalities
Tell whether the ordered pair is a solution of
the inequality.
(3, 1); y > x – 4
y>x−4
1
3–4
1> –1
Substitute (3, 1) for (x, y).

(3, 1) is a solution.
A linear inequality describes a region of
a coordinate plane called a half-plane.
All points in the region are solutions of
the linear inequality. The boundary line
of the region is the graph of the related equation.
Graphing Linear Inequalities
Step 1
Solve the inequality for y (slopeintercept form).
Step 2
Graph the boundary line. Use a solid line
for ≤ or ≥. Use a dashed line for < or >.
Shade the half-plane above the line for y >
Step 3 or ≥. Shade the half-plane below the line
for y < or y ≤. Check your answer.
Example 3: Graphing Linear
Inequalities in Two
Variables
Graph the solutions of the linear inequality.
y  2x – 3
Step 1 The inequality is
already solved for y.
Step 2 Graph the
boundary line y = 2x – 3.
Use a solid line for .
Step 3 The inequality is ,
so shade below the line.
Example 4: Graphing Linear
Inequalities in Two
Variables
Graph the solutions of the linear inequality.
5x + 2y > –8
Step 1 Solve the inequality for y.
5x + 2y > –8
–5x
–5x
2y > –5x – 8
y>
x–4
Step 2 Graph the boundary
line y =
x – 4. Use a dashed line for >.
Example 4 Continued
Graph the solutions of the linear inequality.
5x + 2y > –8
Step 3 The inequality is >, so
shade above the line.
Example 5: Application
Ada has at most 285 beads to make jewelry. A
necklace requires 40 beads, and a bracelet
requires 15 beads.
Let x represent the number of necklaces and y the
number of bracelets.
Write an inequality. Use ≤ for “at most.”
Example 5 Continued
Necklace
beads
40x
plus
bracelet
beads
is at
most
285
beads.
+
15y
≤
285
Solve the inequality for y.
40x + 15y ≤ 285
–40x
–40x
15y ≤ –40x + 285
Subtract 40x from
both sides.
Divide both sides by 15.
Example 5 continued
b. Graph the solutions.
Step 1 Since Ada cannot make a
negative amount of jewelry, the
system is graphed only in
Quadrant I. Graph the boundary
line
for ≤.
=
. Use a solid line
Example 5 Continued
b. Graph the solutions.
Step 2 Shade below the line. Ada
can only make whole numbers of
jewelry. All points on or below the
line with whole number
coordinates are the different
combinations of bracelets and
necklaces that Ada can make.
Try these…
1. You can spend at most $12.00
for drinks at a picnic. Iced tea
costs $1.50 a gallon, and
lemonade costs $2.00 per
gallon. Write an inequality to
describe the situation. Graph
the solutions, describe
reasonable solutions, and then
give two possible
combinations of drinks you
could buy.
1.50x + 2.00y ≤ 12.00
Try these…
2. Write an inequality to represent the graph.