Manuela solved the equation $3-2|0.5 x+1.5|=2$ for one solution. Her work is shown below.
$3-2|0.5 x+1.5|=2$
$-2|0.5 x+1.5|=-1$
$|0.5 x+1.5|=0.5$
$0.5 x+1.5=0.5$
$0.5 x=-1$
$x=-2$
What is the other solution?
A. $x=-6$
B. $x=-4$
C. $x=2$
D. $x=4
Answer (2)
The given equation is 3 − 2∣0.5 x + 1.5∣ = 2 , and one solution is x = − 2 .
To find the other solution, solve the absolute value equation ∣0.5 x + 1.5∣ = 0.5 by considering the case 0.5 x + 1.5 = − 0.5 .
Subtract 1.5 from both sides: 0.5 x = − 2 .
Divide by 0.5 to find the other solution: x = − 4 . Therefore, the other solution is − 4 .
Explanation
Analyzing the Problem 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. The absolute value equation ∣0.5 x + 1.5∣ = 0.5 has two cases: 0.5 x + 1.5 = 0.5 and 0.5 x + 1.5 = − 0.5 . The first case leads to the solution x = − 2 , which is already given. We need to solve the second case.
Isolating the Variable Term We need to solve the equation 0.5 x + 1.5 = − 0.5 for x . First, subtract 1.5 from both sides of the equation: 0.5 x + 1.5 − 1.5 = − 0.5 − 1.5 This simplifies to: 0.5 x = − 2
Solving for x Now, divide both sides of the equation by 0.5 to solve for x : 0.5 0.5 x = 0.5 − 2 This gives us: x = − 4
Finding the Other Solution Therefore, the other solution to the equation is x = − 4 .
Examples
Absolute value equations are useful in various real-world scenarios, such as calculating distances or tolerances in engineering and manufacturing. For example, if you're designing a component that needs to fit within a certain tolerance, you might use an absolute value equation to determine the acceptable range of measurements. Suppose a machine part should be 5mm long, with a tolerance of 0.1mm. The length x must satisfy |x - 5| <= 0.1, which means the length can be between 4.9mm and 5.1mm.
The other solution to the equation 3 − 2∣0.5 x + 1.5∣ = 2 is found by solving the absolute value case ∣0.5 x + 1.5∣ = − 0.5 , leading to the result x = − 4 . Therefore, the correct choice is option B: x = − 4 .
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