Questions: Manuela solved the equation (3-20.5 x+1.5=2) for one solution. Her work is shown below. [ 3-20.5 x+1.5 =2 -20.5 x+1.5 =-1 0.5 x+1.5 =0.5 0.5 x+1.5 =0.5 0.5 x =-1 x =-2 ] What is the other solution to the equation? (x=-6) (x=-4) (x=2) (x=4)

Manuela solved the equation (3-20.5 x+1.5=2) for one solution. Her work is shown below.

[
3-20.5 x+1.5 =2
-20.5 x+1.5 =-1
0.5 x+1.5 =0.5
0.5 x+1.5 =0.5
0.5 x =-1
x =-2
]

What is the other solution to the equation?
(x=-6)
(x=-4)
(x=2)
(x=4)

Transcript text: Manuela solved the equation $3-2|0.5 x+1.5|=2$ for one solution. Her work is shown below. \[ \begin{aligned} 3-2|0.5 x+1.5| & =2 \\ -2|0.5 x+1.5| & =-1 \\ |0.5 x+1.5| & =0.5 \\ 0.5 x+1.5 & =0.5 \\ 0.5 x & =-1 \\ x & =-2 \end{aligned} \] What is the other solution to the equation? $x=-6$ $x=-4$ $x=2$ $x=4$

Solution

Solution Steps

Step 1: Set Up the Equation

We start with the equation given by Manuela: \[ 3 - 2|0.5x + 1.5| = 2 \]

Step 2: Simplify the Equation

Rearranging the equation, we isolate the absolute value: \[ -2|0.5x + 1.5| = -1 \] Dividing both sides by -2 gives: \[ |0.5x + 1.5| = 0.5 \]

Step 3: Solve the Absolute Value Equation

The absolute value equation $ |0.5x + 1.5| = 0.5 $ leads to two cases:

Case 1: \[ 0.5x + 1.5 = 0.5 \] Solving this gives: \[ 0.5x = -1 \implies x = -2 \]
Case 2: \[ 0.5x + 1.5 = -0.5 \] Solving this gives: \[ 0.5x = -2 \implies x = -4 \]

Step 4: List All Solutions

The solutions to the equation are: \[ x = -2 \quad \text{and} \quad x = -4 \]