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Warm-Up Exercises
1. Solve |x – 6| = 4.
2. Solve |x + 5| – 8 = 2.
ANSWER 2, 10
ANSWER –15, 5
3. A frame will hold photographs that are 5 inches by 8inches with an absolute derivation of 0.25 inch forlength and width. What are the minimum andmaximum dimensions for photos?
ANSWERmin: 4.75 in. by 7.75 in.; max: 5.25 in. by 8.25 in.
Warm-Up6.4 Students will be able to solve inequalities that have two variables.
Warm-Up Exercises
Homework Review
Warm-Up ExercisesDaily Homework Quiz For use after Lesson 6.5
Solve the equation. If possible.
ANSWER – 9, 17
ANSWER no solutions
1. x– 4 = 13
2. x + 2 + 7 = 3
3. 2x– 6 + 4 = 20 ANSWER – 5, 11
Warm-Up Exercises
SOLUTION
EXAMPLE 1 Standardized Test Practice
Which ordered pair is not a solution of x – 3y ≤ 6?
A (0, 0) B (6, – 1) C (10, 3) D (– 1, 2)
Check whether each ordered pair is a solution of the inequality.
Test (0, 0): x – 3y ≤ 6
0 – 3(0) ≤ 6
Write inequality.
Substitute 0 for x and 0 for y.
Simplify.0 ≤ 6
Warm-Up ExercisesEXAMPLE 1 Standardized Test Practice
Test (6, – 1): x – 3y ≤ 6
6 – 3(– 1) ≤ 6 Substitute 6 for x and – 1 for y.
Write inequality.
Simplify.
So, (0, 0) is a solution of x – 3y ≤ 6 but (6, – 1) is not a solution.
ANSWER
The correct answer is B. A B C D
9 ≤ 6
Warm-Up ExercisesGUIDED PRACTICE for Example 1
SOLUTION
Tell whether the ordered pair is a solution of – x + 2y < 8.
Check whether each ordered pair is a solution of the inequality.
Test (0, 0 ) – x + 2y < 8.
0 + 2(0) < 8
Write inequality.
Substitute 0 for x and 0 for y.
Simplify.
1. (0, 0)
0 < 8
Warm-Up ExercisesGUIDED PRACTICE for Example 1
ANSWER
So, (0, 0) is a solution of – x + 2y < 8.
Warm-Up Exercises
SOLUTION
GUIDED PRACTICE for Example 1
2. (0, 4)
Check whether each ordered pair is a solution of the inequality.
Test (0, 4 ) – x + 2y < 8.
0 + 2(4) < 8
Write inequality.
Substitute 0 for x and 4 for y.
Simplify.0 + 8 < 8
ANSWER
So, (0, 4) is not a solution of – x + 2y < 8.
Warm-Up ExercisesGUIDED PRACTICE for Example 1
SOLUTION
3. (3, 5)
Check whether each ordered pair is a solution of the inequality.
Test (3, 5 ) – x + 2y < 8.
– 3 + 2(5) < 8
Write inequality.
Substitute – 3 for x and 5 for y.
Simplify.7 < 8
ANSWER
So, (0, 0) is a solution of – x + 2y < 8.
Warm-Up ExercisesEXAMPLE 2 Graph a linear inequality in two variables
Graph the inequality y > 4x – 3.
SOLUTION
Graph the equation y = 4x – 3. The inequality is >, so use a dashed line.
STEP 1
STEP 2
0 > 4(0) – 3?
Test (0, 0) in y > 4x – 3.
0 >–3
Warm-Up ExercisesEXAMPLE 2 Graph a linear inequality in two variables
Shade the half-plane that contains (0, 0), because (0, 0) is a solution of the inequality.
STEP 3
Warm-Up ExercisesEXAMPLE 3 Graph a linear inequality in two variables
Graph the inequality x + 2y ≤ 0
SOLUTION
STEP 1Graph the equation x + 2y = 0. The inequality is < , so use a solid line.
STEP 2
Test (1, 0) in x + 2y ≤ 0
1 ≤ 0
1 + 2(0) ≤ 0?
Warm-Up ExercisesEXAMPLE 3 Graph a linear inequality in two variables
Shade the half-plane that does not contain (1, 0), because (1, 0) is not a solution of the inequality.
STEP 3
Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3
4. Graph the inequality x + 3y ≥ –1
SOLUTION
STEP 1
Graph the equation x + 3y = –1. The inequality is < , so use a solid line.
STEP 2
Test (1, 0) in x + 3y ≤ –1
1 + 2(0) ≤ –1?
1 ≤ –1
Warm-Up ExercisesGUIDED PRACTICE for Examples 2 and 3
STEP 3
Shade the half-plane that contain (1, 0), because (1, 0) is a solution of the inequality.
Warm-Up ExercisesEXAMPLE 4 Graph a linear inequality in one variables
Graph the inequality y > – 3.
SOLUTION
Graph the equation y = – 3. The inequality is >, so use a solid line.
STEP 1
STEP 2
Test (2, 0) in y > – 3.
You substitute only the y-coordinate, because the inequality does not have the variable x.
0 >–3
Warm-Up ExercisesEXAMPLE 4 Graph a linear inequality in one variables
Shade the half-plane that contains (2, 0), because (2, 0) is a solution of the inequality.
STEP 3
Warm-Up ExercisesEXAMPLE 5 Graph a linear inequality in one variables
Graph the inequality x < – 1.
SOLUTION
Graph the equation x = – 1. The inequality is <, so use a dashed line.
STEP 1
STEP 2
Test (3, 0) in x < – 1.
You substitute only the x-coordinate, because the inequality does not have the variable y.
3 <–1
Warm-Up ExercisesEXAMPLE 5 Graph a linear inequality in one variables
Shade the half-plane that does not contains 3, 0), because (3, 0) is not a solution of the inequality.
STEP 3
Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5
5. Graph the inequality y > 1.
SOLUTION
Graph the equation y = 1. The inequality is <, so use a dashed line.
STEP 1
STEP 2
You substitute only the y-coordinate, because the inequality does not have the variable x.
Test (1, 0) in y < 1.
1> 1
Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5
STEP 3
Shade the half-plane that contains (1, 0), because (1, 0) is a solution of the inequality.
Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5
SOLUTION
Graph the equation y = 3. The inequality is <, so use a dashed line.
STEP 1
STEP 2
You substitute only the y-coordinate, because the inequality does not have the variable x.
Test (3, 0) in y < 3.
3> 3
6. Graph the inequality y < 3.
Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5
STEP 3
Shade the half-plane that contains (3, 0), because (3, 0) is a solution of the inequality.
Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5
SOLUTION
Graph the equation y = –2. The inequality is <, so use a dashed line.
STEP 1
STEP 2
You substitute only the y-coordinate, because the inequality does not have the variable x.
Test (2, 0) in y < – 2 .
7. Graph the inequality x < – 2.
2 <–2
Warm-Up ExercisesGUIDED PRACTICE for Examples 4 and 5
STEP 3
Shade the half-plane that does not contains (2, 0), because (2, 0) is not a solution of the inequality.
Warm-Up ExercisesEXAMPLE 6Solve a multi-step problem
Job Earnings
You have two summer jobs at a youth center. You earn $8 per hour teaching basketball and $10 per hour teaching swimming. Let x represent the amount of time (in hours) you teach basketball each week, and let y represent the amount of time (in hours) you teach swimming each week. Your goal is to earn at least $200 per week.
Warm-Up ExercisesEXAMPLE 6Solve a multi-step problem
• Write an inequality that describes your goal in terms of x and y.
• Graph the inequality.
• Give three possible combinations of hours that will allow you to meet your goal.
SOLUTION
Write a verbal model. Then write an inequality.
STEP 1
Warm-Up ExercisesEXAMPLE 6Solve a multi-step problem
STEP 2
Graph the inequality 8x + 10y ≥ 200
First, graph the equation 8x + 10y = 200 in Quadrant I. The inequality is ≥ , so use a solid line.
Warm-Up ExercisesEXAMPLE 6Solve a multi-step problem
Next, test (5, 5) in 8x + 10y ≥ 200
8(5) + 10(5) ≥ 200
90 ≥ 200
Finally, shade the part of Quadrant I that does not contain (5, 5), because (5, 5) is not a solution of the inequality.
STEP 3Choose three points on the graph, such as (13, 12), (14, 10), and (16, 9). The table shows the total earnings for each combination of hours.
Warm-Up ExercisesEXAMPLE 6Solve a multi-step problem
Warm-Up ExercisesGUIDED PRACTICE for Example 6
8. WHAT IF? In Example 6, suppose that next summer you earn $9 per hour teaching basketball and $12.50 per hour teaching swimming. Write and graph an inequality that describes your goal. Then give three possible combinations of hours that will help you meet your goal.
SOLUTION
Write a verbal model. Then write an inequality.
STEP 1
Warm-Up ExercisesGUIDED PRACTICE for Example 6
STEP 2
Graph the inequality 9x + 12.5y ≥ 200
First, graph the equation 9x + 12.5y = 200 in Quadrant I. The inequality is ≥ , so use a solid line.
Warm-Up ExercisesGUIDED PRACTICE for Example 6
Next, test (5, 5) in 9x + 12.5y ≥ 200 9(5) + 12.5(5) ≥ 200
107.5 ≥ 200
Finally, shade the part of Quadrant I that does not contain (5, 5), because (5, 5) is not a solution of the inequality.
Warm-Up ExercisesGUIDED PRACTICE for Example 6
STEP 3
Choose three points on the graph, such as (8, 12), (12, 10), and (16, 9). The table shows the total earnings for each combination of hours.
