Question

1. a rectangle has a perimeter of 76 cm. the length measures (x + 4). the width measures (x+2). what is the value of x, in centimeters?
2. what is(are) the solution(s) to the equation?

|8x| = 0
a. no solution
b. x = 0 and x = 8
c. x = 0 only
d. x = 8 only
7.what is(are) the solution(s) to the equation?
|2x - 5| + 3 = 12
a. no solution
b. x = -7 and x = 2
c. x = -2 and x = 7
d. x = 7 only

1. what is (are) the solution(s) to the equation?

|7x - 30| = -9
a. no solution
b. x = -3 and x = 3
c. x = -3 only
d. x = 3 only

Explanation:

Step1: Solve equation \(|8x| = 0\)

The absolute value of a number \(|a|\) is defined as \(|a|=

$$\begin{cases}a, & a\geq0\\-a, & a < 0\end{cases}$$

\). For \(|8x| = 0\), we know that if \(|a| = 0\), then \(a = 0\). So \(8x=0\).

Step2: Solve for \(x\)

Divide both sides of the equation \(8x = 0\) by \(8\). Using the formula \(\frac{8x}{8}=\frac{0}{8}\).

Step3: Solve equation \(|2x - 5|+3 = 12\)

First, isolate the absolute - value term. Subtract \(3\) from both sides: \(|2x - 5|=12 - 3=9\).
Then, use the property \(|a|=b\) (\(b\geq0\)) implies \(a = b\) or \(a=-b\). So \(2x-5 = 9\) or \(2x - 5=-9\).
For \(2x-5 = 9\), add \(5\) to both sides: \(2x=9 + 5=14\), then divide by \(2\): \(x = 7\).
For \(2x-5=-9\), add \(5\) to both sides: \(2x=-9 + 5=-4\), then divide by \(2\): \(x=-2\).

Step4: Analyze equation \(|7x - 30|=-9\)

The absolute value of a number \(|a|\) is always non - negative (\(|a|\geq0\) for all real \(a\)). Since \(-9<0\), the equation \(|7x - 30|=-9\) has no solution.

Answer:

1. C. \(x = 0\) only
2. C. \(x=-2\) and \(x = 7\)
3. A. No solution