Question

Determine the solutions of the equation:

Absolute value of the quantity one fourth times x plus 7 end quantity minus 3 equals 24.

a x = −136 and x = 136
b x = −136 and x = 80
c x = −112 and x = 80
d x = −80 and x = 80

Answer 1

To solve the equation

\[
\left| \frac{1}{4}x + 7 \right| - 3 = 24,
\]

we start by isolating the absolute value:

\[
\left| \frac{1}{4}x + 7 \right| = 24 + 3 = 27.
\]

Next, we can split this into two cases based on the definition of absolute value.

**Case 1:**

\[
\frac{1}{4}x + 7 = 27.
\]

To solve for \(x\), we subtract 7 from both sides:

\[
\frac{1}{4}x = 27 - 7 = 20.
\]

Multiplying both sides by 4 gives:

\[
x = 20 \times 4 = 80.
\]

**Case 2:**

\[
\frac{1}{4}x + 7 = -27.
\]

Again, we subtract 7 from both sides:

\[
\frac{1}{4}x = -27 - 7 = -34.
\]

Multiplying both sides by 4 gives:

\[
x = -34 \times 4 = -136.
\]

Thus, the solutions to the equation are

\[
x = 80 \quad \text{and} \quad x = -136.
\]

The correct answer is

\[
\boxed{b} \quad x = -136 \text{ and } x = 80.
\]