Problem 22 - Algebra for College Students

Question

Solve each equation. $$ |0.3 x|=9 $$

Step-by-Step Solution

Answer

The solutions are x = 30 and x = -30.

Step 1: Understand the Absolute Value Equation

The absolute value equation is given as \(|0.3x| = 9\). The absolute value \(|a|\) is equal to the value of \(|a|\) itself if \(|a|\) is positive and \(-a\) if \(|a|\) is negative.

Step 2: Set Up Two Equations

Since we have an absolute value, write two separate equations: \(|0.3x| = 9\) implies \(0.3x = 9\) and \(0.3x = -9\).

Step 3: Solve the First Equation

Solve \(0.3x = 9\) by dividing both sides by 0.3: \(\frac{0.3x}{0.3} = \frac{9}{0.3}\). This gives \(x = 30\).

Step 4: Solve the Second Equation

Solve \(0.3x = -9\) by dividing both sides by 0.3: \(\frac{0.3x}{0.3} = \frac{-9}{0.3}\). This gives \(x = -30\).

Step 5: State the Solutions

The solutions to the equation \(|0.3x| = 9\) are \(x = 30\) and \(x = -30\).

Key Concepts

absolute value

The absolute value of a number is its distance from zero on the number line. It is always non-negative. For any real number \(a\), the absolute value is denoted by \(|a|\).
For example:

• If \(a = 5\), then \(|a| = 5\).
• If \(a = -5\), then \(|a| = 5\).

This means that both positive and negative numbers have the same absolute value as they are the same distance from zero.
Understanding absolute value is crucial for solving equations involving absolute values, as it requires considering both the positive and negative solutions.

equation solving steps

Solving absolute value equations involves several clear steps.
Let's use the example \(|0.3x| = 9\) from the exercise:

• **Step 1: Understand the equation**
  We start with the absolute value equation \(|0.3x| = 9\).
• **Step 2: Break into two cases**
  The equation implies two separate cases: \(\text{Case 1: } 0.3x = 9\) and \(\text{Case 2: } 0.3x = -9\).
• **Step 3: Solve each equation individually**
  For \(\text{Case 1: } 0.3x = 9\), divide both sides by 0.3 to get \(x = 30\). For \(\text{Case 2: } 0.3x = -9\), divide both sides by 0.3 to get \(x = -30\).
• **Step 4: Combine the solutions**
  The solutions to the equation \(|0.3x| = 9\) are \(x = 30\) and \(x = -30\).

By following these steps, we ensure that all possible solutions to the absolute value equation are considered.

positive and negative solutions

When dealing with absolute value equations, it's important to consider both positive and negative solutions.
Why? Because the definition of absolute value covers both positive and negative outcomes for the same result.

• For \(|a| = b\), where \(b\) is a positive number, both \((a = b)\) and \((a = -b)\) are valid.

In our exercise \(|0.3x| = 9\), we considered:

• Case 1: When \(0.3x = 9\).
• Case 2: When \(0.3x = -9\).

By solving both cases:

• We found \(x = 30\).
• We also found \(x = -30\).

That is why there are two solutions for \(|0.3x| = 9\): \(x = 30\) and \(x = -30\). Considering both cases ensures we don't miss any possible solutions.

linear equations

Linear equations are equations of the first degree. This means the variable is only to the power of one. For example, \(\text{y = mx + b}\) is a linear equation where \(m\) and \(b\) are constants.

In the context of absolute value equations, once we break them down into individual cases, what remains are linear equations which are straightforward to solve.
For the equation \(|0.3x| = 9\), we broke it down to:

• Case 1: \(0.3x = 9\).
• Case 2: \(0.3x = -9\).

Both of these are simple linear equations. Solving them involves basic algebra:

• For \(0.3x = 9\), divide by 0.3 to get \(x = 30\).
• For \(0.3x = -9\), divide by 0.3 to get \(x = -30\).

Thus, understanding linear equations facilitates solving absolute value equations efficiently.