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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

Half-plane 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 half-plane containing the point if the ordered pair is a solution of the inequality. Shade the other half-plane 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 half-plane 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 x-coordinate because the inequality does not have the variable y . 0 4 x

Step 3 Shade the half-plane 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