Solve each equation.
[tex]|3 x+9|=30[/tex]
Answer (2)
The solutions to the equation ∣3 x + 9∣ = 30 are x = 7 and x = − 13 . To obtain these solutions, we solved two cases: one for the positive value and one for the negative value of the expression inside the absolute value. By following these steps, we can ensure that all potential solutions are found.
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We are given the absolute value equation ∣3 x + 9∣ = 30 .
We consider two cases: 3 x + 9 = 30 and 3 x + 9 = − 30 .
Solving the first case, we get x = 7 .
Solving the second case, we get x = − 13 .
The solutions are x = 7 and x = − 13 , so the answer is x = 7 , − 13 .
Explanation
Understanding the problem We are given the absolute value equation ∣3 x + 9∣ = 30 and asked to solve for x . Absolute value equations can be solved by considering two cases.
Solving Case 1 Case 1: 3 x + 9 = 30 . We solve for x by first subtracting 9 from both sides of the equation: 3 x + 9 − 9 = 30 − 9 3 x = 21 Then, we divide both sides by 3: 3 3 x = 3 21 x = 7
Solving Case 2 Case 2: 3 x + 9 = − 30 . We solve for x by first subtracting 9 from both sides of the equation: 3 x + 9 − 9 = − 30 − 9 3 x = − 39 Then, we divide both sides by 3: 3 3 x = 3 − 39 x = − 13
Final Answer Therefore, the solutions to the equation ∣3 x + 9∣ = 30 are x = 7 and x = − 13 .
Examples
Absolute value equations are useful in many real-world scenarios. For example, when manufacturing parts, there is often a tolerance for the dimensions. If a part is supposed to be 5 cm long, but it can be off by up to 0.1 cm, the actual length x must satisfy the equation ∣ x − 5∣ ≤ 0.1 . Solving this inequality tells us the acceptable range of lengths for the part.
