Solve $\left|-\frac{x}{3}\right|=8$ for $x$.
Answer (2)
Rewrite the equation: − 3 x = 8 becomes 3 1 ∣ x ∣ = 8 .
Isolate the absolute value: Multiply both sides by 3 to get ∣ x ∣ = 24 .
Solve for x : Consider both positive and negative values.
The solutions are x = − 24 and x = 24 , so the answer is x = − 24 , 24 .
Explanation
Understanding the Problem We are given the equation − 3 x = 8 and asked to solve for x . This equation involves an absolute value, which means we need to consider both positive and negative possibilities for the expression inside the absolute value.
Simplifying the Equation First, we can rewrite the equation as 3 − x = 8 . Using the property that ∣ ab ∣ = ∣ a ∣∣ b ∣ , we can rewrite the equation as − 3 1 ∣ x ∣ = 8 , which simplifies to 3 1 ∣ x ∣ = 8 .
Isolating the Absolute Value Next, we multiply both sides of the equation by 3 to isolate the absolute value term: 3 × 3 1 ∣ x ∣ = 3 × 8 , which gives us ∣ x ∣ = 24 .
Solving for x Now, we solve for x by considering both positive and negative values. If ∣ x ∣ = 24 , then x can be either 24 or -24. Therefore, x = 24 or x = − 24 .
Final Answer The solutions to the equation − 3 x = 8 are x = 24 and x = − 24 .
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 ∣ x − 5∣ ≤ 0.1 , where x is the actual length of the part. Solving this inequality will give you the acceptable range of lengths for the part. Similarly, absolute value can be used to model situations where only the magnitude of a quantity matters, such as the speed of a car, regardless of direction.
To solve the absolute value equation − 3 x = 8 , we isolate the absolute value to get ∣ x ∣ = 24 , leading to two solutions: x = 24 and x = − 24 .
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