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 4 –4 + 1 4 –3 <  Substitute (–2, 4) for (x, y). (–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 (slope-intercept form). Step 2 Graph the boundary line. Use a solid line for ≤ or ≥. Use a dashed line for < or >. Step 3 Shade the half-plane above the line for y > 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 y = x – 4. Step 2 Graph the boundary line 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. Write a linear inequality to describe the situation. 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 bracelet 40x + 15y ≤ plus is at most 285 beads. 40x + 15y ≤ 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 . Use a solid line for ≤.

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.