Solving Linear Inequalities 5-5 Solving Linear Inequalities Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1 Holt Algebra 1

Warm Up Graph each inequality. 1. x > –5 2. y ≤ 0 3. Write –6x + 2y = –4 in slope-intercept form, and graph.

Objective Graph and solve linear inequalities in two variables.

VOCABULARY Linear inequality: similar to a linear equation, but the equal sign is replaced with an inequality symbol. Solution of a linear inequality: any ordered pair that makes the inequality true.

Example 1A: 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 1B: 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.

You try!!!! Tell whether the ordered pair is a solution of the inequality. a. (4, 5); y < x + 1 b. (1, 1); y > x – 7

When the inequality is written as: the points on the boundary line _________________ of the inequality and the line is _____________ When the inequality is written as: the points on the boundary line _________________ of the inequality and the line is _____________ STEP 1 – HOW SHOULD I DRAW THE BOUNDARY LINE? 𝒚≤𝒐𝒓 𝒚≥ 𝒚<𝒐𝒓 𝒚> are solutions are not solutions solid dashed When the inequality is written as: the points _______ the boundary line are _________________ When the inequality is written as: the points _______ the boundary line are _________________ 𝒚>𝒐𝒓 𝒚≥ 𝒚<𝒐𝒓 𝒚≤ STEP 2 – HOW SHOULD I SHADE? above below solutions of the inequality solutions of the inequality

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 2A: 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 2A Continued Graph the solutions of the linear inequality. y  2x – 3 Substitute (0, 0) for (x, y) because it is not on the boundary line. Check y  2x – 3 0 2(0) – 3 0 –3   A false statement means that the half-plane containing (0, 0) should NOT be shaded. (0, 0) is not one of the solutions, so the graph is shaded correctly.

The point (0, 0) is a good test point to use if it does not lie on the boundary line. Helpful Hint

Example 2B: 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 Use a dashed line for >. y = x – 4.

Example 2B Continued Graph the solutions of the linear inequality. 5x + 2y > –8 Step 3 The inequality is >, so shade above the line.

 Example 2B Continued Graph the solutions of the linear inequality. 5x + 2y > –8 Substitute ( 0, 0) for (x, y) because it is not on the boundary line. Check y > x – 4 0 (0) – 4 0 –4 >  The point (0, 0) satisfies the inequality, so the graph is correctly shaded.

Try on your own!!! Graph the solutions of the linear inequality. 2x – y – 4 > 0

Example 3: 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.”

Necklace beads bracelet Example 3a Continued Necklace beads bracelet 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 3b 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 3b 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.

Example 3c c. Give two combinations of necklaces and bracelets that Ada could make. Two different combinations of jewelry that Ada could make with 285 beads could be 2 necklaces and 8 bracelets or 5 necklaces and 3 bracelets. (2, 8) (5, 3) 

Example Together Write an inequality to represent the graph. y-intercept: 1; slope: Write an equation in slope-intercept form. The graph is shaded above a dashed boundary line. Replace = with > to write the inequality

Try on your own!!! Write an inequality to represent the graph. y-intercept: slope:

HOMEWORK PG. 364-366 #12-21, 30-40(evens), 41, 42