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Chapter 3: Problem 28

Solve. $$|5 x+4|+2=5$$

Short Answer

Expert verified
The solutions are \(x = -\frac{1}{5}\) and \(x = -\frac{7}{5}\).

Step by step solution

01

Isolate the Absolute Value Expression

First, isolate the absolute value expression by subtracting 2 from both sides of the equation: \(|5x + 4| + 2 = 5\) \(|5x + 4| = 3\)
02

Set Up Two Equations

The absolute value equation \(|5x + 4| = 3\) has two possible solutions. Set up two separate equations to account for both scenarios: \(5x + 4 = 3\) and \(5x + 4 = -3\)
03

Solve the First Equation

Solve the first equation \(5x + 4 = 3\) by isolating x: \(5x + 4 - 4 = 3 - 4\) \(5x = -1\) \(x = -\frac{1}{5}\)
04

Solve the Second Equation

Solve the second equation \(5x + 4 = -3\) by isolating x: \(5x + 4 - 4 = -3 - 4\) \(5x = -7\) \(x = -\frac{7}{5}\)
05

Combine the Solutions

Combine the solutions from both equations: The solutions are \(x = -\frac{1}{5}\) and \(x = -\frac{7}{5}\).

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Key Concepts

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

Understanding Absolute Value
The absolute value of a number refers to its distance from zero on the number line, regardless of direction. Hence, it is always a non-negative value. For example, the absolute value of both 3 and -3 is 3, written as \(|3| = 3\) and \(|-3| = 3\). This concept helps in representing both positive and negative scenarios in a single expression.
Grasping Linear Equations
Linear equations form the backbone of algebra. They appear as a straight line when graphed on a coordinate plane. The general form is \(ax + b = c\), where 'a,' 'b,' and 'c' are constants. Solving linear equations involves isolating the variable (often represented as x) by performing inverse operations to both sides.
Step-by-Step Solution for Absolute Value Equations
Let's dive into solving absolute value equations using a clear step-by-step approach, as demonstrated in the provided exercise.

To solve \(|5x + 4| + 2 = 5\):
  • Start by isolating the absolute value expression. Subtract 2 from both sides: \(|5x + 4| = 3\).
  • Next, set up two separate linear equations to account for the two possible cases: \(5x + 4 = 3\) and \(5x + 4 = -3\).
  • Then, solve these equations independently. For \(5x + 4 = 3\), isolate x by simplifying: \(5x = -1\) and \(x = -\frac{1}{5}\).
  • Similarly, solve \(5x + 4 = -3\): \(5x = -7\) and \(x = -\frac{7}{5}\).

Combining these solutions, we find that the values of x that satisfy the original equation are \(x = -\frac{1}{5}\) and \(x = -\frac{7}{5}\). This method ensures you consider all possible solutions within the absolute value framework.

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