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Chapter 1: Problem 92

Solve each equation. $$ |7 x+12|=|x-8| $$

Short Answer

Expert verified
The solutions to the equation are \(x = -\frac{10}{3}\) and \(x = -\frac{1}{2}\).

Step by step solution

01

- Understand the Absolute Value Equation

The equation involves absolute values, so we need to consider two cases for each side: one where the expressions inside the absolute values are positive and one where they are negative.
02

- Set Up the Two Cases

Since \(|a| = |b|\) implies \(a = b\) or \(a = -b\), set up the two equations: 1. \(7x + 12 = x - 8\)2. \(7x + 12 = -(x - 8)\)
03

- Solve the First Case

For the equation \(7x + 12 = x - 8\): Subtract \(x\) from both sides to get \(6x + 12 = -8\). Next, subtract 12 from both sides: \(6x = -20\). Finally, divide by 6: \(x = -\frac{10}{3}\).
04

- Solve the Second Case

For the equation \(7x + 12 = -(x - 8)\): Distribute the negative sign on the right side: \(7x + 12 = -x + 8\). Add \(x\) to both sides: \(8x + 12 = 8\). Next, subtract 12 from both sides: \(8x = -4\). Finally, divide by 8: \(x = -\frac{1}{2}\).
05

- Verify Solutions

Plug both found values \(-\frac{10}{3}\) and \(-\frac{1}{2}\) back into the original equation to ensure they satisfy \(|7x + 12| = |x - 8|\). Both should satisfy the original equation.

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Key Concepts

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

Understanding Absolute Value
Absolute value essentially measures the distance of a number from zero on the number line. It is always non-negative, meaning it鈥檚 either positive or zero. In equations involving absolute values, like \(|7x + 12| = |x - 8|\), we consider both the positive and negative scenarios because absolute value \( |a|\) can equal \(a\) or \(-a\). To solve such equations, break them into two separate cases. One where the expressions inside the absolute values are equal, and another where one is the negative of the other.
Algebraic Equations
Algebraic equations involve variables and constants. Solving them means finding the value of the variable that makes the equation true. Consider the equation \(7x + 12 = x - 8\). Solve for \(x\) to find the solution. Here鈥檚 how to solve it step-by-step:
  • First, subtract \(x\) from both sides: \(6x + 12 = -8\)
  • Next, subtract \12\ from both sides: \(6x = -20\)
  • Finally, divide by \6\: \(x = -\frac{10}{3}\)
This straightforward approach helps you isolate the variable and simplifies the equation step-by-step.
Linear Equations
Linear equations are the simplest type of algebraic equations. They represent a straight line when graphed on a coordinate plane. The general form is \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. In our example, after considering both scenarios:
  • From \(7x + 12 = x - 8\), we found \(x = -\frac{10}{3}\)
  • From \(7x + 12 = -(x - 8)\), after simplifying, we get \(x = -\frac{1}{2}\)
Both solutions should satisfy the original absolute value equation when substituted back into the expression, thereby confirming their validity.

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