Question

lve the equation. graph the solution(s), if possible.

1. $2|7y - 8| = -28$
2. $\frac{|4z + 8|}{-3} = -4$

Explanation:

Problem 6: Solve \( 2|7y - 8| = -28 \)

Step 1: Isolate the absolute value expression

Divide both sides of the equation by 2:
\( |7y - 8| = \frac{-28}{2} \)
\( |7y - 8| = -14 \)

Step 2: Analyze the absolute value

The absolute value of any real number is always non - negative (i.e., \( |a| \geq 0 \) for all real numbers \( a \)). Here, the right - hand side of the equation \( |7y - 8|=-14 \) is negative. Since a non - negative number (the absolute value) cannot be equal to a negative number, there is no solution for this equation.

Step 1: Isolate the absolute value expression

Multiply both sides of the equation by - 3 to get rid of the denominator on the left - hand side:
\( |4z + 8|=(-4)\times(-3) \)
\( |4z + 8| = 12 \)

Step 2: Set up two cases for the absolute value equation

Case 1: The expression inside the absolute value is equal to the positive value on the right - hand side.
\( 4z+8 = 12 \)
Subtract 8 from both sides: \( 4z=12 - 8 \)
\( 4z = 4 \)
Divide both sides by 4: \( z=\frac{4}{4}=1 \)

Case 2: The expression inside the absolute value is equal to the negative value on the right - hand side.
\( 4z + 8=-12 \)
Subtract 8 from both sides: \( 4z=-12 - 8 \)
\( 4z=-20 \)
Divide both sides by 4: \( z=\frac{-20}{4}=-5 \)

Answer:

No solution

Problem 7: Solve \( \frac{|4z + 8|}{-3}=-4 \)