- Simplify the equation to isolate the absolute value: \(|0.5x + 1.5| = 0.5\).
- Apply the property \(|A| = k \iff A = k \text{ or } A = -k\) to set up two cases.
- Solving the first case \(0.5x + 1.5 = 0.5\) gives \(x = -2\).
- Solving the second case \(0.5x + 1.5 = -0.5\) results in the second solution \(x = -4\).
- Final Answer: \(\boxed{-4}\)
Explanation
- Analysis of the Given Equation
The equation provided is \(3 - 2|0.5x + 1.5| = 2\). By subtracting 3 from both sides, we get \(-2|0.5x + 1.5| = -1\). Dividing both sides by -2 yields the simplified form \(|0.5x + 1.5| = 0.5\). This absolute value expression describes the distance of \(0.5x + 1.5\) from 0 on the number line, which is equal to 0.5.
- Applying the Absolute Value Property
The definition of the absolute value function states that \(|A| = k\) is equivalent to two separate equations: \(A = k\) and \(A = -k\). In this problem, we have \(A = 0.5x + 1.5\) and \(k = 0.5\). This gives us two cases:
Case 1: \(0.5x + 1.5 = 0.5\)
Case 2: \(0.5x + 1.5 = -0.5\)
- Solving for the Second Variable Value
We know from the problem statement that solving the first case gives \(x = -2\). Now, let us solve the second case:
\[0.5x + 1.5 = -0.5\]
Subtracting 1.5 from both sides of the equation, we obtain:
\[0.5x = -0.5 - 1.5\]
\[0.5x = -2\]
Finally, dividing both sides by 0.5:
\[x = \frac{-2}{0.5} = -4\]
Thus, the other solution is \(x = -4\).
Examples
Absolute value equations are widely used in engineering and physics to define tolerances. For example, if a machine component must have a length of \(10\) cm with a tolerance of \(0.5\) cm, the acceptable lengths \(L\) can be expressed as \(|L - 10| \le 0.5\). By solving the equality \(|L - 10| = 0.5\), we find the boundary conditions for the component's length, which are \(L = 9.5\) cm and \(L = 10.5\) cm. Understanding how to handle these equations ensures that manufacturing processes stay within safe and functional limits.