Answered

Select the correct answer.

What is the solution to $4|x-3|+1=1$ ?


A$x=3$
B$x=3$ or $x=4$
C$x=3$ or $x=6$
DNo solutions exist.

Verified Answer :

  • Isolate the absolute value term by subtracting 1 from both sides: \(4|x - 3| = 0\).
  • Simplify by dividing by 4: \(|x - 3| = 0\).
  • Apply the property that if \(|A| = 0\), then \(A = 0\), which gives \(x - 3 = 0\).
  • Solve the resulting linear equation to get the final answer: \(x = 3\).
  • The correct option is \(\boxed{x = 3}\).

Explanation

  1. Isolating the absolute value term

To solve the equation \(4|x - 3| + 1 = 1\), we first isolate the term containing the absolute value. Subtracting 1 from both sides of the equation yields:
\[4|x - 3| = 1 - 1\]
\[4|x - 3| = 0\]

  1. Simplifying the equation

Next, we divide both sides of the equation by 4 to further isolate the absolute value expression:
\[|x - 3| = \frac{0}{4}\]
\[|x - 3| = 0\]

  1. Solving for x

The absolute value of a number, \(|A|\), is equal to 0 if and only if the expression inside the absolute value is equal to 0. Therefore, we set the expression inside the absolute value, \(x - 3\), to 0:
\[x - 3 = 0\]
Adding 3 to both sides gives the solution:
\[x = 3\]

  1. Selecting the correct option

We have found that the only solution to the equation is \(x = 3\). Comparing this to the given options, we select the correct answer:

  • \(x = 3\)
  • \(x = 3\) or \(x = 4\)
  • \(x = 3\) or \(x = 6\)
  • No solutions exist.

The correct option is \(x = 3\).

Examples

Absolute value equations are used in various fields to define distances and margins of error. For example, if a machine in a factory is calibrated to produce a part with a length of exactly \(3\) inches, but it allows for a variation, we might express the acceptable range using an absolute value inequality or equation. If the manufacturer states that the deviation from 3 inches must be zero (meaning the part must be exactly 3 inches), the equation \(|x - 3| = 0\) helps identify that only parts with a length of \(x = 3\) satisfy this strict requirement. Understanding these equations helps in quality control and precise engineering.

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