Mastering Absolute Values and Piecewise Functions

1. A temperature reading is represented by the absolute value equation |x - 30| < 5. What is the range of temperatures that are considered acceptable?

• 20 < x < 30

• 15 < x < 25

• 30 < x < 40

• 25 < x < 35

2. The distance from a point on a number line is given by the inequality |x + 4| > 3. Solve for x and interpret the solution in terms of distance from -4.

• x = -4

• x < -7 or x > -1

• x < -5 or x > 0

• -7 < x < -1

3. A piecewise function is defined as f(x) = { 2x + 1, x < 0; 3x - 2, x >= 0 }. Evaluate f(-3) and f(2). What do these values represent in a real-world context?

• f(-3) = -5, f(2) = 4

• f(-3) = 1, f(2) = 6

• f(-3) = -2, f(2) = 5

• f(-3) = -4, f(2) = 3

4. The absolute value inequality |2x - 6| ≤ 8 represents a range of values for x. Solve the inequality and explain what this range means in a practical scenario.

• The range of values for x is from -1 to 7, inclusive.

• x < -1

• x > 7

• -1 < x < 7

5. A car's speed is modeled by the function v(t) = |t - 60|, where t is the time in seconds. What is the speed of the car at t = 70 seconds?

• 20

• 5

• 10

• 15

6. The height of a ball thrown into the air can be modeled by the piecewise function h(t) = { -16t^2 + 64t, 0 ≤ t ≤ 4; 0, t > 4 }. What is the maximum height of the ball?

• 32

• 64

• 48

• 80

7. Solve the absolute value equation |3x + 1| = 7. What do the solutions represent in a real-world context, such as profit or loss?

• x = 0 and x = -5

• x = 2 and x = -\frac{8}{3}

• x = 3 and x = -2

• x = 1 and x = -3

8. A store has a sale where the price of an item is represented by the function p(x) = { 50 - |x - 20|, 0 ≤ x ≤ 40; 0, otherwise }. What is the price of the item when x = 15?

• 40

• 50

• 30

• 45

9. The inequality |x - 10| < 4 represents a range of values for x. Solve the inequality and describe a situation where this range might apply, such as acceptable weights for a package.

• The range of values for x is (0, 20). This could represent acceptable weights for a package that should be between 0 and 20 pounds.

• The range of values for x is (8, 12). This could represent acceptable weights for a package that should be between 8 and 12 pounds.

• The range of values for x is (5, 15). This could represent acceptable weights for a package that should be between 5 and 15 pounds.

• The range of values for x is (6, 14). This could represent acceptable weights for a package that should be between 6 and 14 pounds.

10. A piecewise function models the cost of a service: C(x) = { 20, x < 5; 15x, 5 ≤ x ≤ 10; 100, x > 10 }. What is the cost for 7 services, and how does this reflect the pricing strategy?

• 105

• 120

• 90

• 70