Question

describe the values of c for which the equation has no solution. explain your reasoning.

1. $3x - 5 = 3x - c$
2. $|x - 7| = c$
3. a safety regulation states that the minimum height of a handrail is 30 inches. the maximum height is 38 inches. write an absolute value equation that represents the minimum and maximum heights.
4. the fastest recorded speed in a standing position on a skateboard is about 146.7 kilometers per hour. the fastest recorded speed on inline skates is about 113.6 feet per second. which speed is faster?
5. the perimeter $p$ (in yards) of a soccer field is represented by the formula $p = 2\ell + 2w$, where $\ell$ is the length (in yards) and $w$ is the width (in yards).

a. solve the formula for $w$.
b. the perimeter of the soccer field is 330 yards. find the width of the field.
c. about what percent of the field is inside the circle?
(image of a soccer field with $\ell = 100$ yd and a circle with radius 10 yd)

1. your car needs new brakes. you call a dealership and a local mechanic for prices.

	cost of parts	labor cost per hour
local mechanic	$45	$89

a. after how many hours are the total costs the same at both places? justify your answer.
b. when do the repairs cost less at the dealership? at the local mechanic? explain.

1. the speed of light is 11,176,920 miles per minute and the distance from earth to the sun at a specific time is 92,955,807 miles. about how long does it take for sunlight to reach earth?
2. consider the equation $|4x + 20| = 6x$. without solving, how do you know that $x = -2$ is an extraneous solution?

Explanation:

Problem 7

Step1: Simplify the equation

Subtract $3x$ from both sides:
$3x - 5 - 3x = 3x - c - 3x$
$-5 = -c$

Step2: Analyze no-solution condition

For the equation to have no solution, the simplified form must be a false statement. If $c
eq 5$, then $-5
eq -c$, which is a contradiction with no solution.

Step1: Recall absolute value property

The absolute value of any real number is non-negative: $|x-7| \geq 0$ for all real $x$.

Step2: Identify no-solution condition

If $c < 0$, there is no real number $x$ such that $|x-7| = c$, since a non-negative value cannot equal a negative number.

Step1: Find midpoint of heights

Calculate the average of min and max:
$\frac{30 + 38}{2} = 34$

Step2: Find distance from midpoint

Subtract midpoint from max:
$38 - 34 = 4$

Step3: Write absolute value equation

Let $h$ = handrail height. The equation is $|h - 34| = 4$.

Answer:

The equation has no solution when $c
eq 5$.

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Problem 8