Given the function $f(x)=-0.5|2 x+2|+1$, for what values of $x$ is $f(x)=6$?

There are no values of x for which the function f ( x ) = 6 , as the absolute value cannot equal a negative number. Thus, there is no solution to the equation. Therefore, the answer is that there are no values of x that meet the condition.
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Answered by Anonymous | 2025-07-04

Set f ( x ) = 6 , resulting in the equation − 0.5∣2 x + 2∣ + 1 = 6 .
Isolate the absolute value term: − 0.5∣2 x + 2∣ = 5 .
Divide by -0.5: ∣2 x + 2∣ = − 10 .
Since the absolute value cannot be negative, there is no solution: no solution ​ .

Explanation

Problem Analysis We are given the function f ( x ) = − 0.5∣2 x + 2∣ + 1 and we want to find the values of x for which f ( x ) = 6 .

Isolating the Absolute Value We need to solve the equation − 0.5∣2 x + 2∣ + 1 = 6 for x . Let's isolate the absolute value term. First, subtract 1 from both sides: − 0.5∣2 x + 2∣ = 6 − 1 − 0.5∣2 x + 2∣ = 5

Dividing by -0.5 Now, divide both sides by -0.5: ∣2 x + 2∣ = − 0.5 5 ​ ∣2 x + 2∣ = − 10

Analyzing the Result The absolute value of any real number is always non-negative. Therefore, ∣2 x + 2∣ cannot be equal to -10. This means there is no solution to the equation f ( x ) = 6 .

Final Answer Therefore, there is no value of x for which f ( x ) = 6 .


Examples
Absolute value functions are used in many real-world applications, such as calculating distances or tolerances in engineering. For example, when manufacturing parts, engineers use absolute values to specify the acceptable deviation from a target measurement. If a part is supposed to be 10 cm long, a tolerance of ± 0.1 cm can be expressed using an absolute value inequality: ∣ x − 10∣ ≤ 0.1 , where x is the actual length of the part. This ensures that the part's length is within the acceptable range.

Answered by GinnyAnswer | 2025-07-04