Problem 44 - Precalculus Student Solutions Manual 5th

Question

Solve each equation or inequality. $$|8-3 x|-3=-2$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{7}{3}\) and \(x = 3\).

Step 1: Isolate the Absolute Value Expression

Add 3 to both sides of the equation to isolate the absolute value expression:\(|8-3x|-3+3 = -2+3\)This simplifies to:\(|8-3x| = 1\)

Step 2: Set Up Two Separate Equations

Since the value inside an absolute value expression can be either positive or negative, set up two separate equations:1. \(8-3x = 1\)2. \(8-3x = -1\)

Step 3: Solve the First Equation

Solve the equation \(8-3x = 1\):Subtract 8 from both sides:\(-3x = 1 - 8\)Simplify:\(-3x = -7\)Divide by -3:\(x = \frac{7}{3}\)

Step 4: Solve the Second Equation

Solve the equation \(8-3x = -1\):Subtract 8 from both sides:\(-3x = -1 - 8\)Simplify:\(-3x = -9\)Divide by -3:\(x = 3\)

Step 5: Verify the Solutions

Plugging both values back into the original equation to verify:For \(x = \frac{7}{3}\): \(|8 - 3(\frac{7}{3})| = |8-7| = 1\) and \(|8-7|-3 = 1-3= -2\) [True]For \(x = 3\):\(|8 - 3(3)| = |8-9| = 1\) and \(|8-9|-3 = 1-3 = -2\) [True]

Key Concepts

Absolute Value

Absolute value can be thought of as the distance a number is from zero, regardless of its direction on the number line. In other words, it strips a number of any negative sign.
For example, the absolute value of both -7 and 7 is 7, since both are 7 units away from zero.
The notation for absolute value is two vertical bars around the number or expression, like this: \(|x|\).
This can yield two possible solutions when included in an equation, since the expression inside the bars can be either positive or negative and still contribute the same absolute value.

Equation Solving

Solving equations involving absolute values involves a few straightforward steps:

• Isolate the absolute value expression.
• Set up the two possible equations: one for the positive scenario and one for the negative scenario.
• Solve both equations separately.
• Verify your solutions by substituting them back into the original equation.

In our example, we had \(|8-3x|-3 = -2\).
First, we added 3 to both sides to isolate the absolute value expression, resulting in \(|8-3x| = 1\).
This gave us two potential equations to solve: \((8-3x = 1)\) and \((8-3x = -1)\).
We then solved each equation separately to find solutions for \((x = \frac{7}{3})\) and \((x = 3)\).
Finally, substituting these values back into the original equation confirmed their validity.

Precalculus

Precalculus often includes fundamental topics like functions, equations, and inequalities that pave the way for calculus. A strong grasp of these concepts is essential.
Absolute value equations are a key topic in precalculus because they teach students how to handle non-linear problems and prepare them for more complex scenarios in calculus.
Other important areas include polynomial and rational functions, trigonometry, and limits.
Understanding how to manipulate and solve equations, including absolute value equations, ensures a solid foundation as you advance in your math studies.