Solve the equation $|5 x+2|-3=1$
$x=$ $\square$
Answer (1)
Isolate the absolute value term: ∣5 x + 2∣ = 4 .
Split the equation into two cases: 5 x + 2 = 4 and 5 x + 2 = − 4 .
Solve for x in each case: x = 5 2 and x = − 5 6 .
The solutions are x = 5 2 and x = − 5 6 , so the final answer is − 5 6 , 5 2 .
Explanation
Understanding the Problem We are given the equation ∣5 x + 2∣ − 3 = 1 and we need to solve for x . This equation involves an absolute value, which means we need to consider two cases.
Isolating the Absolute Value First, we isolate the absolute value term by adding 3 to both sides of the equation: ∣5 x + 2∣ − 3 + 3 = 1 + 3 ∣5 x + 2∣ = 4
Case 1: Positive Value Now we consider two cases:
Case 1: The expression inside the absolute value is equal to 4. 5 x + 2 = 4
Solving for x (Case 1) Solving for x in Case 1, we subtract 2 from both sides: 5 x + 2 − 2 = 4 − 2 5 x = 2
Solution for x (Case 1) Then, we divide by 5: x = 5 2
Case 2: Negative Value Case 2: The expression inside the absolute value is equal to -4. 5 x + 2 = − 4
Solving for x (Case 2) Solving for x in Case 2, we subtract 2 from both sides: 5 x + 2 − 2 = − 4 − 2 5 x = − 6
Solution for x (Case 2) Then, we divide by 5: x = − 5 6
Final Answer Therefore, the solutions to the equation are x = 5 2 and x = − 5 6 .
Examples
Absolute value equations are useful in many real-world scenarios, such as calculating distances or tolerances in engineering and manufacturing. For example, if a machine part needs to be within a certain tolerance of a specified length, an absolute value equation can be used to determine the acceptable range of lengths. Suppose a metal rod is designed to be 10 cm long, but it can be off by at most 0.05 cm. The length x of the rod must satisfy the equation ∣ x − 10∣ ≤ 0.05 . Solving this inequality gives the acceptable range of lengths for the metal rod.
