Question

practice and problem - solving exercises
mathematical practices
a practice
solve each equation. graph and check your solutions.
see problem 1.

1. |b| = \frac{1}{2}
2. 4 = |y|
3. |n| + 3 = 7
4. 7 = |s| - 3
5. |x| - 10 = -2
6. 5|d| = 20
7. -3|m| = -9
8. |y| + 3 = 3

solve each equation. if there is no solution, write no solution.
see problems 2 and 3.

1. |r - 8| = 5
2. |c + 4| = 6
3. 2 = |g + 3|
4. 3 = |m + 2|
5. -2|7d| = 14
6. -3|2w| = -12
7. 3|v - 3| = 9
8. 2|d + 4| = 8
9. |4f + 1| - 2 = 5
10. |3t - 2| + 6 = 2
11. 4|2y - 3| - 1 = 11
12. 3|x + 2| + 4 = 13
13. -4|k| = 12
14. | - 3n| - 2 = 4
15. -4|k + 1| = 16

solve and graph each inequality.
see problems 4 and 5.

1. |x| \geq 3
2. |x| < 5
3. |x + 3| < 5
4. |y + 8| \geq 3
5. |y - 2| \leq 1
6. |p - 7| \leq 3
7. |2c - 5| < 9
8. |3t + 1| > 8
9. |4w + 1| > 11
10. |5t - 4| \geq 16
11. |4x + 7| > 19
12. |2v - 1| \leq 9
13. |3d - 7| > 28
14. |2f + 9| \leq 13
15. |5m - 9| \geq 24
16. quality control the ideal length of one type of model airplane is 90 cm. the actual length may vary from ideal by at most 0.05 cm. what are the acceptable lengths for the model airplane?
17. basketball the ideal circumference of a womens basketball is 28.75 in. the actual circumference may vary from the ideal by at most 0.25 in. what are the acceptable circumferences for a womens basketball?

solve each equation or inequality. if there is no solution, write no solution.

1. |2d| + 3 = 21
2. 1.2|5p| = 3.6
3. \left|d + \frac{1}{2}\

ight| + \frac{3}{4} = 0

1. |f| - \frac{2}{3} = \frac{5}{6}
2. 3|5y - 7| - 6 = 24
3. |t| + 2.7 = 4.5
4. -2|c - 4| = -8.4
5. \frac{|y|}{-3} = 5
6. |n| - \frac{5}{4} < 5
7. \frac{7}{8} < |c + 7|
8. 4 - 3|m + 2| > -14
9. | - 3d| \geq 6.3
10. think about a plan the monthly average temperature t for san francisco, california, is usually within 7.5°f of 56.5°f, inclusive. what is the monthly average temperature in san francisco?

• should you model this situation with an equation or an inequality?
• how can you use the given information to write the equation or inequality?

Explanation:

To solve the problem, we'll address each part:

Part 1: Model with Equation or Inequality?

The problem states the temperature \( T \) is "within \( 7.5^\circ\text{F} \) of \( 56.5^\circ\text{F} \), inclusive." The phrase "within... of" and "inclusive" indicates a range of values, so we use an inequality (specifically, an absolute value inequality) to model this.

Part 2: Writing the Inequality

The absolute value of the difference between \( T \) and \( 56.5 \) must be less than or equal to \( 7.5 \) (since "within \( 7.5 \)" means the distance from \( 56.5 \) is at most \( 7.5 \), and "inclusive" means equality is allowed).

Mathematically, this is:

$$ |T - 56.5| \leq 7.5 $$

Solving the Inequality (Optional, but to find the range)

To solve \( |T - 56.5| \leq 7.5 \), we use the property of absolute value inequalities:
If \( |x| \leq a \) (where \( a \geq 0 \)), then \( -a \leq x \leq a \).

Applying this:

$$ -7.5 \leq T - 56.5 \leq 7.5 $$

Add \( 56.5 \) to all parts:

$$ 56.5 - 7.5 \leq T \leq 56.5 + 7.5 $$

Simplify:

$$ 49 \leq T \leq 64 $$

Final Answers

• Equation or Inequality? Inequality (because we model a range of temperatures, not a single value).
• Writing the Inequality: Use the absolute value of the difference between \( T \) and \( 56.5 \), set less than or equal to \( 7.5 \): \( |T - 56.5| \leq 7.5 \).

For the monthly average temperature range:
The acceptable temperatures are between \( 49^\circ\text{F} \) and \( 64^\circ\text{F} \), inclusive.

Answer:

To solve the problem, we'll address each part:

Part 1: Model with Equation or Inequality?

The problem states the temperature \( T \) is "within \( 7.5^\circ\text{F} \) of \( 56.5^\circ\text{F} \), inclusive." The phrase "within... of" and "inclusive" indicates a range of values, so we use an inequality (specifically, an absolute value inequality) to model this.

Part 2: Writing the Inequality

The absolute value of the difference between \( T \) and \( 56.5 \) must be less than or equal to \( 7.5 \) (since "within \( 7.5 \)" means the distance from \( 56.5 \) is at most \( 7.5 \), and "inclusive" means equality is allowed).

Mathematically, this is:

$$ |T - 56.5| \leq 7.5 $$

Solving the Inequality (Optional, but to find the range)

To solve \( |T - 56.5| \leq 7.5 \), we use the property of absolute value inequalities:
If \( |x| \leq a \) (where \( a \geq 0 \)), then \( -a \leq x \leq a \).

Applying this:

$$ -7.5 \leq T - 56.5 \leq 7.5 $$

Add \( 56.5 \) to all parts:

$$ 56.5 - 7.5 \leq T \leq 56.5 + 7.5 $$

Simplify:

$$ 49 \leq T \leq 64 $$

Final Answers

• Equation or Inequality? Inequality (because we model a range of temperatures, not a single value).
• Writing the Inequality: Use the absolute value of the difference between \( T \) and \( 56.5 \), set less than or equal to \( 7.5 \): \( |T - 56.5| \leq 7.5 \).

For the monthly average temperature range:
The acceptable temperatures are between \( 49^\circ\text{F} \) and \( 64^\circ\text{F} \), inclusive.