Solve [tex]\frac{2}{3}|3 x+6|=12[/tex] for [tex]x[/tex].
Answer (2)
Multiply both sides by 2 3 to get ∣3 x + 6∣ = 18 .
Split into two cases: 3 x + 6 = 18 and 3 x + 6 = − 18 .
Solve 3 x + 6 = 18 to get x = 4 .
Solve 3 x + 6 = − 18 to get x = − 8 . The solutions are x = − 8 , 4 .
Explanation
Problem Analysis We are given the equation 3 2 ∣3 x + 6∣ = 12 and we want to solve for x .
Isolating the Absolute Value First, we multiply both sides of the equation by 2 3 to isolate the absolute value term: 2 3 ⋅ 3 2 ∣3 x + 6∣ = 2 3 ⋅ 12 ∣3 x + 6∣ = 18
Considering Two Cases Now we consider the two cases for the absolute value. Case 1: 3 x + 6 = 18 and Case 2: 3 x + 6 = − 18 .
Solving Case 1 For Case 1, we solve for x : 3 x + 6 = 18 3 x = 18 − 6 3 x = 12 x = 3 12 x = 4
Solving Case 2 For Case 2, we solve for x : 3 x + 6 = − 18 3 x = − 18 − 6 3 x = − 24 x = 3 − 24 x = − 8
Final Answer Therefore, the solutions are x = 4 and x = − 8 .
Examples
Absolute value equations are useful in many real-world scenarios, such as calculating distances or tolerances in engineering and manufacturing. For example, if you are designing a machine part that needs to be within a certain tolerance of a specific size, you can use absolute value equations to determine the acceptable range of sizes for the part. Suppose a machine part should be 5 cm long, with a tolerance of 0.01 cm. This means the actual length can be between 4.99 cm and 5.01 cm. The equation representing this situation is ∣ x − 5∣ = 0.01 , where x is the actual length of the part. Solving this equation gives the acceptable range for the length of the machine part.
The solutions to the equation 3 2 ∣3 x + 6∣ = 12 are x = 4 and x = − 8 . After isolating the absolute value and considering two cases, both solutions were derived. This process is a standard method for solving absolute value equations in mathematics.
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