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Chapter 3: Problem 27

Solve. $$|4 x-3|+1=7$$

Short Answer

Expert verified
The solutions are \(x = \frac{9}{4}\) and \(x = \frac{-3}{4}\).

Step by step solution

01

Isolate the absolute value expression

Start by isolating the absolute value expression on one side of the equation. Subtract 1 from both sides of the equation.\[ |4x - 3| + 1 = 7 \]Subtract 1:\[ |4x - 3| = 6 \]
02

Set up two separate equations

To solve for the absolute value expression, set up two separate equations to account for both the positive and negative scenarios:\[ 4x - 3 = 6 \]\[ 4x - 3 = -6 \]
03

Solve the first equation

Solve the first equation by isolating \(x\):\[ 4x - 3 = 6 \]Add 3 to both sides:\[ 4x = 9 \]Divide both sides by 4:\[ x = \frac{9}{4} \]
04

Solve the second equation

Solve the second equation by isolating \(x\):\[ 4x - 3 = -6 \]Add 3 to both sides:\[ 4x = -3 \]Divide both sides by 4:\[ x = \frac{-3}{4} \]
05

Combine the solutions

Combine the solutions from both equations to get the final answer. The possible values of \(x\) are \(\frac{9}{4}\) and \(\frac{-3}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value
Absolute value is a concept that denotes the distance of a number from zero on the number line, without considering its direction. It is always a non-negative value.
For any real number, the absolute value is represented with vertical bars, like this: \(|x|\). For instance, \(|3| = 3\) and \(|-3| = 3\).
The absolute value of a number can be described as:
  • \(|x| = x \ if \ x \) is greater than or equal to zero.
  • \(|x| = -x \ if \ x \) is less than zero.
This means that when you take the absolute value of a negative number, you

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