Solve the equation $|2x + 4| = 12$. The solutions are: $\square$
Answer (2)
Split the absolute value equation into two cases: 2 x + 4 = 12 and 2 x + 4 = − 12 .
Solve the first equation 2 x + 4 = 12 to get x = 4 .
Solve the second equation 2 x + 4 = − 12 to get x = − 8 .
The solutions are x = 4 and x = − 8 , so the final answer is 4 , − 8 .
Explanation
Understanding the problem We are given the absolute value equation ∣2 x + 4∣ = 12 . To solve this, we need to consider two cases. The first case is when the expression inside the absolute value is positive or zero, and the second case is when the expression inside the absolute value is negative.
Solving for x (Case 1) Case 1: 2 x + 4 = 12 . We solve for x by subtracting 4 from both sides of the equation: 2 x + 4 − 4 = 12 − 4 2 x = 8 Then, we divide both sides by 2: 2 2 x = 2 8 x = 4
Solving for x (Case 2) Case 2: 2 x + 4 = − 12 . We solve for x by subtracting 4 from both sides of the equation: 2 x + 4 − 4 = − 12 − 4 2 x = − 16 Then, we divide both sides by 2: 2 2 x = 2 − 16 x = − 8
Final Answer Therefore, the solutions to the equation ∣2 x + 4∣ = 12 are x = 4 and x = − 8 .
Examples
Absolute value equations are useful in many real-world scenarios. For example, in manufacturing, if you need to produce parts that are a certain size, say 10 cm, but you allow for a tolerance of 0.1 cm, then the actual size, x , of the part must satisfy the equation ∣ x − 10∣ ≤ 0.1 . This means the part can be between 9.9 cm and 10.1 cm. Similarly, in physics, absolute values are used to describe the magnitude of a vector, such as velocity or force, without regard to direction.
The solutions to the equation ∣2 x + 4∣ = 12 are x = 4 and x = − 8 . This is derived from considering two cases based on the definition of absolute value. Thus, we can express the final answer as 4 , − 8 .
;
