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Chapter 1: Problem 9

Find the real solutions, if any, of each equation. $$ |3 x|=15 $$

Short Answer

Expert verified
x = 5 and x = -5

Step by step solution

01

Understand the Absolute Value Equation

The equation provided is an absolute value equation: \( |3x| = 15 \). Absolute value equations mean the expression inside the absolute value can be equal to the positive or negative value of the number on the other side.
02

Set Up Two Separate Equations

Since the absolute value of a number equals 15, the number itself can either be 15 or -15. Thus, we set up two separate equations: \( 3x = 15 \) and \( 3x = -15 \).
03

Solve for x in Each Equation

Now solve each equation separately: \[ 3x = 15 \] \[ x = \frac{15}{3} = 5 \] \[ 3x = -15 \] \[ x = \frac{-15}{3} = -5 \]
04

Write the Final Solutions

The solutions for the equation \( |3x| = 15 \) are \( x = 5 \) and \( x = -5 \). Thus, the real solutions are \( x = 5 \) and \( x = -5 \).

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Key Concepts

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

Solving Equations
Solving equations is one of the core skills in mathematics. It's like finding the missing piece of a puzzle. In an equation, you have expressions on either side of an equal sign. The goal is to find the value of the variable that makes the equation true.

In the given exercise, we solve the equation involving an absolute value: \( |3x| = 15 \).

Here鈥檚 a quick rundown on solving equations, step-by-step:
  • First, understand what the equation represents.
  • Next, manipulate the expressions to isolate the variable.

Breaking down complex equations into simpler parts makes the process smoother. For absolute value equations, this usually means setting up two separate equations once the absolute value is removed.
Absolute Value Properties
Absolute value measures the distance of a number from zero on the number line, regardless of direction. It鈥檚 always non-negative.

So when solving \( |3x| = 15 \), you鈥檙e dealing with the distance from zero. This can be 15 or -15. That gives us two situations to consider:
  • \( 3x = 15 \)
  • \( 3x = -15 \)

Here are a few key properties of absolute values to keep in mind:
  • \( |a| \) is always non-negative.
  • \( |a| = a \) if \( a \) is positive or zero.
  • \( |a| = -a \) if \( a \) is negative.

These properties guide us in setting up separate equations to solve.
Real Solutions
On solving \( 3x = 15 \) and \( 3x = -15 \), we get:
  • \( x = 5 \)
  • \( x = -5 \)

Both these solutions make the original absolute value equation true. Therefore, these are considered real solutions.
Real solutions are the actual values of the variable that satisfy the given equation.
Solving for real solutions involves:
  • Checking each possible value.
  • Verifying that it satisfies the original equation.

Understanding these steps makes solving absolute value equations easier and more logical.

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