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?
A. $x=-6$
B. $x=-4$
C. $x=2$
D. $x=4$
Answer (1)
Isolate the absolute value term: ∣0.5 x + 1.5∣ = 0.5 .
Consider two cases: 0.5 x + 1.5 = 0.5 and 0.5 x + 1.5 = − 0.5 .
Solve the first case to find x = − 2 .
Solve the second case to find the other solution: x = − 4 .
Explanation
Problem Analysis We are given the equation 3 − 2∣0.5 x + 1.5∣ = 2 and one solution x = − 2 . We need to find the other solution.
Isolating the Absolute Value First, we isolate the absolute value term. Subtract 3 from both sides of the equation: 3 − 2∣0.5 x + 1.5∣ − 3 = 2 − 3 − 2∣0.5 x + 1.5∣ = − 1
Simplifying the Equation Next, divide both sides by -2: − 2 − 2∣0.5 x + 1.5∣ = − 2 − 1 ∣0.5 x + 1.5∣ = 0.5
Considering Two Cases Now, we consider the two cases for the absolute value: Case 1: 0.5 x + 1.5 = 0.5 Case 2: 0.5 x + 1.5 = − 0.5
Solving Case 1 Solve Case 1: 0.5 x + 1.5 = 0.5 Subtract 1.5 from both sides: 0.5 x = 0.5 − 1.5 0.5 x = − 1 Divide by 0.5: x = 0.5 − 1 x = − 2 This is the solution Manuela found.
Solving Case 2 Solve Case 2: 0.5 x + 1.5 = − 0.5 Subtract 1.5 from both sides: 0.5 x = − 0.5 − 1.5 0.5 x = − 2 Divide by 0.5: x = 0.5 − 2 x = − 4
Finding the Other Solution Therefore, the other solution to the equation is x = − 4 .
Examples
Absolute value equations are useful in many real-world scenarios, such as calculating distances or tolerances in engineering. For example, if you are manufacturing a part that needs to be exactly 5 cm long, but you allow for a tolerance of 0.1 cm, you can express this as an absolute value equation: ∣ x − 5∣ = 0.1 , where x is the actual length of the part. Solving this equation will give you the acceptable range of lengths for the part.
