Savanah solved the equation [tex]$3+4\left|\frac{x}{2}+3\right|=11$[/tex] for one solution. Her work is shown below:
1. [tex]$4\left|\frac{x}{2}+3\right|=8$[/tex]
2. [tex]$\left|\frac{x}{2}+3\right|=2$[/tex]
3. [tex]$\frac{x}{2}+3=2$[/tex]
4. [tex]$\frac{x}{2}=-1$[/tex]
5. [tex]$x=-2$[/tex]
What is the other solution to the given absolute value equation?
[tex]$x=$[/tex]
Answer (2)
Isolate the absolute value term: 2 x + 3 = 2 .
Split the absolute value equation into two cases: 2 x + 3 = 2 and 2 x + 3 = − 2 .
Solve the first case (already done by Savanah) to find x = − 2 .
Solve the second case to find the other solution: x = − 10 .
The other solution is − 10 .
Explanation
Problem Analysis We are given the absolute value equation 3 + 4 2 x + 3 = 11 and we know that one solution to this equation is x = − 2 . Our goal is to find the other solution.
Isolating the Absolute Value First, we need to isolate the absolute value term. We can do this by subtracting 3 from both sides of the equation:
3 + 4 2 x + 3 − 3 = 11 − 3
4 2 x + 3 = 8
Next, we divide both sides of the equation by 4:
4 4 2 x + 3 = 4 8
2 x + 3 = 2
Considering Both Cases Now, we consider the two possible cases for the absolute value. Either the expression inside the absolute value is equal to 2, or it is equal to -2. This gives us two equations to solve:
Case 1: 2 x + 3 = 2
Case 2: 2 x + 3 = − 2
Verifying the Given Solution We are given that Savanah already solved Case 1, which is 2 x + 3 = 2 . Let's quickly solve it to confirm that we get x = − 2 :
2 x + 3 = 2
Subtract 3 from both sides:
2 x = 2 − 3
2 x = − 1
Multiply both sides by 2:
x = − 2
So, x = − 2 is indeed one solution.
Finding the Other Solution Now, let's solve Case 2, which is 2 x + 3 = − 2 :
2 x + 3 = − 2
Subtract 3 from both sides:
2 x = − 2 − 3
2 x = − 5
Multiply both sides by 2:
x = − 10
Therefore, the other solution to the absolute value equation is x = − 10 .
Final Answer The other solution to the given absolute value equation is x = − 10 .
Examples
Absolute value equations are useful in many real-world scenarios, such as calculating errors in measurements. For example, if you are manufacturing parts that need to be a certain length, the absolute value can help you determine how much the actual length can deviate from the desired length while still being acceptable. Suppose a machine is set to cut metal rods to 10 cm, and the tolerance is 0.5 cm. This means the length can be between 9.5 cm and 10.5 cm. The equation representing this situation is $|x - 10|
\leq 0.5 , w h ere x$ is the actual length of the rod. Solving absolute value equations and inequalities helps ensure quality control in manufacturing and other fields.
The other solution to the absolute value equation is x = − 10 . This was found by solving the second case after isolating the absolute value. Savanah solved one case, and we determined the alternate case to arrive at this solution.
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