Question

in exercises 1–3, solve the equation and check your answer.

1. 1 - 3x = -17
2. 11x + 2x = 19x
3. 10(x - 1) = -2x + 62
4. a furnace repair person charges an initial fee of $80 plus $30 per hour to do repairs

a. after how many hours would the cost of the repair be at least $320?
b. how many hours did the repair person work if the total bill was $230?
in exercises 5–7, solve the equation. graph the solution(s), if possible.

1. |3x + 9| = 18
2. 2|7y - 8| = -28
3. \\(\frac{|4z + 8|}{-3} = -4\\)

in exercises 8–10, solve the inequality. graph the solution.

1. 2 - 3x ≥ -x + 8
2. 4t - 7 ≥ 25
3. 6x - x + 10 < 9 - 4

in exercises 11–13, solve the inequality. graph the solution, if possible.

1. |14x + 7| < 35
2. |7w - 2| + 8 ≤ -9
3. -2|4 + 2x| < -20
4. the ideal width of a certain conveyor belt for a manufacturing plant is 50 inches. the width of an actual conveyor belt can vary from the ideal width by at most \\(\frac{7}{32}\\) of an inch.

a. write an absolute value inequality to describe this situation.
b. solve the inequality to find the acceptable widths, in inches, for this conveyor belt.
in exercises 15 and 16, determine whether the relation is a function. explain.

1. (-3, 2), (0, 3), (3, 4), (6, 5), (-3, 6)
2. (3, 1), (-5, 1), (-3, -1), (-5, -1), (1, ...)
3. the equation y = 1725 - 75x represents the amount of money y (in dollars) left in your lunch account after x weeks.

a. identify the independent and dependent variables.
b. twenty weeks go by. find the domain and range of the function.
exercises 18–20: graph the linear function.

1. f(x) = -x - 1
2. y(x) = \\(\frac{5}{3}x\\)
3. g(x) = 4 - \\(\frac{1}{2}x\\)

Explanation:

Let's solve problem 3: \( 10(x - 1)=-2x + 62 \)

Step 1: Expand the left side

Use the distributive property \( a(b - c)=ab - ac \). Here, \( a = 10 \), \( b=x \), \( c = 1 \), so \( 10(x - 1)=10x-10 \). The equation becomes \( 10x-10=-2x + 62 \)

Step 2: Add \( 2x \) to both sides

To get all \( x \) terms on one side, add \( 2x \) to both sides: \( 10x+2x-10=-2x+2x + 62 \), which simplifies to \( 12x-10 = 62 \)

Step 3: Add 10 to both sides

To isolate the term with \( x \), add 10 to both sides: \( 12x-10 + 10=62 + 10 \), so \( 12x=72 \)

Step 4: Divide by 12

Divide both sides by 12: \( \frac{12x}{12}=\frac{72}{12} \), which gives \( x = 6 \)

Answer:

\( x = 6 \)