
Graph Linear Inequalities in Two Variables
VOCABULARY
Linear inequality in two variables The result of replacing the 5 sign in a
linear equation with <,≤,> or ≥
Solution of an inequality in two variables An ordered pair (x, y) that
produces a true statement when the values are substituted into an inequality in
two variables x and y
Graph of an inequality in two variables The set of points that represent all
solutions of the inequality
Halfplane Either half of the coordinate plane that is divided by the
boundary line of a linear inequality
Example 1 Check solutions of a linear inequality
Tell whether the ordered pair is a solution of 3x  4y >
9.
a. (2, 0) b. (2, 21)
Solution
a.
3x  4y > 9 Write inequality. 3( 2 )  4( 0 )
9 Substitute 2 for x and 0 for
y. 6 > 9 x Simplify. (2, 0) is not a solution of 3x  4y > 9.
b.
3x  4y > 9 Write inequality. 3( 2 )  4( 1 )
9 Substitute 2 for x and 1
for y. 10 > 9 ✓ Simplify. (2, 1) is a solution of 3x  4y > 9.
GRAPHING A LINEAR INEQUALITY IN TWO VARIABLES
Step 1 Graph the boundary line. Use a dashed line
for < or >, and use a solid line for ≤or ≥.
Step 2 Test a point not on the boundary line by
checking whether the ordered pair is a solution of the inequality.
Step 3 Shade the halfplane containing the point if
the ordered pair is a solution of the inequality. Shade the
other halfplane if the ordered pair is not a solution.
Example 2 Graph a linear inequality in two variables
Graph the inequality .
Solution
Step 1 Graph the equation . The
inequality is <, so use a dashed line.
Step 2 Test (0, 0) in .
0 <(0)+4
0 < 4 ✓
Step 3 Shade the halfplane that contains (0, 0)
because (0, 0) is a solution of the inequality.
Example 3 Graph a linear inequality in one variable
Graph the inequality x ≥ 4.
Solution
Step 1 Graph the equation x 5 4. The inequality is ≥, so use a
solid line.
Step 2 Test (0, 3) in x ≥ 4. You only substitute the xcoordinate
because the inequality does not have the variable y . 0 ≥ 4 x
Step 3 Shade the halfplane that does not contain (0, 3), because (0, 3)
is not a solution of the inequality.
Guided Practice Graph the inequality.
1. 2y + 4x > 8
2. y < 2
