Problem 23 - College Algebra with Modeling and Visualization

Question

Solve the absolute value equation. $$|3-4 x|=5$$

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{1}{2}\) and \(x = 2\).

Step 1: Understanding the Absolute Value Equation

We are given the equation \(|3 - 4x| = 5\). The absolute value of a number represents its distance from zero on the number line, regardless of direction. This means \(3 - 4x\) could be either 5 or -5, leading us to two separate equations to solve.

Step 2: Setting Up the First Equation

First, consider the case where the expression inside the absolute value is 5. We set up the equation: \[ 3 - 4x = 5 \]

Step 3: Solving the First Equation

To solve \(3 - 4x = 5\), subtract 3 from both sides to isolate terms involving \(x\): \[ -4x = 2 \]Now, divide by -4: \[ x = -\frac{1}{2} \]

Step 4: Setting Up the Second Equation

Now consider the case where the expression inside the absolute value is -5. We set up the equation: \[ 3 - 4x = -5 \]

Step 5: Solving the Second Equation

To solve \(3 - 4x = -5\), subtract 3 from both sides: \[ -4x = -8 \]Divide by -4: \[ x = 2 \]

Step 6: Verifying the Solutions

Check both solutions by substituting them back into the original equation:For \(x = -\frac{1}{2}\),\(|3 - 4(-\frac{1}{2})| = |3 + 2| = |5| = 5\).For \(x = 2\),\(|3 - 4(2)| = |3 - 8| = |-5| = 5\).Both solutions satisfy the original equation.

Key Concepts

Solving Equations

Solving equations often involves finding the value of the unknown variable that satisfies the equation. In the context of absolute value equations like \(|3-4x|=5\), this means determining the values of \(x\) such that the expression inside the absolute value has a magnitude of 5.

When solving, you break the absolute value equation into two separate cases. This is because the absolute value of a quantity can be either its positive value or its negative value, explaining its distance from zero. Let's see why:

• For \(3-4x = 5\), you solve by isolating \(x\) to find \(x = -\frac{1}{2}\).
• For \(3-4x = -5\), solve similarly and find \(x = 2\).

After solving these two equations independently, you check the solutions by plugging them back into the original equation to ensure they satisfy it. Thus, we learn that solving absolute value equations typically results in two potential solutions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition or multiplication). They form the building blocks of equations.

Let's break down the algebraic expression \(3 - 4x\) from the original absolute value equation.

• "3" is a constant term, which always retains its value.
• "-4x" includes a coefficient (-4) multiplied by a variable (\(x\)).

In the equation \(|3-4x| = 5\), this expression can represent two different linear equations depending on whether \(3-4x\) equals 5 or -5.

Solving these expressions involves operations such as addition, subtraction, division, and multiplication to isolate the variable, uncovering its value. Understanding how these parts interact enables solving the equations accurately.

Distance on Number Line

The concept of distance on a number line is crucial in understanding absolute values. Absolute values measure a number's distance from zero on the number line, making all values non-negative regardless of direction.

For instance, the absolute value \(|3-4x|\) in our equation depicts how far the expression \(3-4x\) is from zero. Absolute values are practical:

• They simplify calculations by focusing on magnitude rather than direction.
• In the equation \(|3-4x|=5\), it indicates the expression’s distance from zero is 5 units.

Hence, whether the expression \(3-4x\) evaluates to 5 or -5, they both represent the same distance from zero, showcasing the symmetrical nature of absolute values in relation to the number line.