Solve $|4-2 x|=|3 x+2|$.
Answer (2)
Split the absolute value equation into two cases: 4 − 2 x = 3 x + 2 and 4 − 2 x = − ( 3 x + 2 ) .
Solve the first case: 4 − 2 x = 3 x + 2 which simplifies to 5 x = 2 , so x = 5 2 .
Solve the second case: 4 − 2 x = − 3 x − 2 which simplifies to x = − 6 .
The solutions are x = 5 2 and x = − 6 , so the final answer is x = − 6 or x = 5 2 .
Explanation
Understanding the Problem We are given the equation ∣4 − 2 x ∣ = ∣3 x + 2∣ . Our goal is to find all values of x that satisfy this equation. The absolute value of a number represents its distance from zero. Therefore, if ∣ a ∣ = ∣ b ∣ , then either a = b or a = − b . This is because both a and − a have the same distance from zero.
Solving the Two Cases We consider two cases:
Case 1: 4 − 2 x = 3 x + 2 . In this case, we solve for x :
4 − 2 x = 3 x + 2 4 − 2 = 3 x + 2 x 2 = 5 x x = 5 2
Case 2: 4 − 2 x = − ( 3 x + 2 ) . In this case, we also solve for x :
4 − 2 x = − 3 x − 2 4 + 2 = − 3 x + 2 x 6 = − x x = − 6
Final Answer Therefore, the solutions to the equation ∣4 − 2 x ∣ = ∣3 x + 2∣ are x = 5 2 and x = − 6 .
Examples
Absolute value equations are useful in many real-world scenarios. For example, in engineering, when designing a bridge, the stress on a material must be within a certain tolerance. This can be modeled using absolute value equations to ensure the stress doesn't deviate too much from the ideal value. Similarly, in manufacturing, absolute value equations can be used to control the dimensions of parts, ensuring they are within acceptable limits. In finance, absolute value can model deviations from expected returns or investment targets, helping to manage risk.
To solve the equation ∣4 − 2 x ∣ = ∣3 x + 2∣ , we break it into two cases: 4 − 2 x = 3 x + 2 and 4 − 2 x = − ( 3 x + 2 ) . This leads to the solutions x = 5 2 and x = − 6 . Both solutions are valid when checked in the original equation.
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