Solve each equation.

[tex]|3x + 9| = 30[/tex]

The solutions to the absolute value equation ∣3 x + 9∣ = 30 are x = 7 and x = − 13 . We solve this by considering two cases based on the definition of absolute value. Each case leads us to a different solution for x .
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Answered by Anonymous | 2025-07-04

Split the absolute value equation into two cases: 3 x + 9 = 30 and 3 x + 9 = − 30 .
Solve the first case: 3 x + 9 = 30 ⇒ 3 x = 21 ⇒ x = 7 .
Solve the second case: 3 x + 9 = − 30 ⇒ 3 x = − 39 ⇒ x = − 13 .
The solutions are x = 7 and x = − 13 , so the final 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 have two possible cases to consider.

Solving Case 1 Case 1: The expression inside the absolute value is equal to 30. So, we have the equation 3 x + 9 = 30 . Subtracting 9 from both sides gives 3 x = 30 − 9 , which simplifies to 3 x = 21 . Dividing both sides by 3 gives x = 3 21 ​ = 7 .

Solving Case 2 Case 2: The expression inside the absolute value is equal to -30. So, we have the equation 3 x + 9 = − 30 . Subtracting 9 from both sides gives 3 x = − 30 − 9 , which simplifies to 3 x = − 39 . Dividing both sides by 3 gives x = 3 − 39 ​ = − 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. Similarly, absolute value equations can be used to model distances, errors, and deviations from a target value.

Answered by GinnyAnswer | 2025-07-04