Problem 14 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\left|\frac{x+2}{2}\right|=7$$

Step-by-Step Solution

Answer

x = 12 or x = -16

Step 1 - Understand absolute value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. Therefore, if \(|A| = B\), it means \((A = B)\) or \((A = -B)\).

Step 2 - Set up two equations

Given \(|\frac{x+2}{2}|=7\), this implies two conditions: \(\frac{x+2}{2}=7\) and \(\frac{x+2}{2}=-7\).

Step 3 - Solve the first equation

Solve \(\frac{x+2}{2}=7\). Multiply both sides by 2 to get \(x + 2 = 14\). Next, subtract 2 from both sides: \((x = 12)\).

Step 4 - Solve the second equation

Solve \(\frac{x+2}{2}=-7\). Multiply both sides by 2 to get \(x + 2 = -14\). Next, subtract 2 from both sides: \((x = -16)\).

Step 5 - State the solutions

The solutions to the equation \(|\frac{x+2}{2}|=7\) are \(x = 12\) and \(x = -16\).

Key Concepts

Understanding Absolute Value

Absolute value is a key concept in mathematics that measures the distance of a number from zero on the number line, without considering direction.
The absolute value is always a non-negative number. For any real number \(x\), the absolute value is denoted as \(|x|\).
For example, \(|3| = 3\) and \(|-3| = 3\), because both 3 and -3 are three units away from zero.
It's crucial to understand that when you solve an absolute value equation like \(|A| = B\), it's equivalent to two separate equations: \A = B\ and \A = -B\. This concept is foundational for solving more complex equations and is often explored in precalculus. When solving absolute value equations, always remember:

• Identify the expression inside the absolute value
• Set up two separate equations
• Solve each equation independently
• Check your solutions by substituting them back into the original equation

Mastering absolute value will help you succeed in advanced topics in mathematics.

Equation Solving

Solving equations is one of the most essential skills in mathematics. When you solve an equation, you find the value(s) of the variable that make the equation true.
To solve the given absolute value equation \(|\frac{x + 2}{2}| = 7\), you can break it down into two separate equations based on the definition of absolute value.
1. Equation 1: \frac{x + 2}{2} = 7\
2. Equation 2: \frac{x + 2}{2} = -7\
Let's solve each of these equations step by step.

Starting with Equation 1:

• Multiply both sides by 2: \(x + 2 = 14\)
• Subtract 2 from both sides: \(x = 12\)

Now solve Equation 2:

• Multiply both sides by 2: \(x + 2 = -14\)
• Subtract 2 from both sides: \(x = -16\)

Therefore, the solutions are \(x = 12\) and \(x = -16\).
Always verify by substituting each solution back into the original equation to ensure they satisfy the equation.

Introduction to Precalculus

Precalculus is a course designed to prepare students for calculus and other higher-level math courses.
It combines skills and concepts from algebra and trigonometry. Being good at solving absolute value equations is one of the many essential skills you'll practice.
In precalculus, you'll often encounter equations and inequalities involving absolute values. These problems strengthen your algebraic skills and improve your problem-solving abilities.
Mastering topics in precalculus not only aids in knowing how to work with functions, polynomials, and other algebraic structures, but also equips you with the strategies to tackle complex calculus problems.
To succeed in precalculus, focus on:

• Understanding the properties of functions
• Mastering trigonometric identities
• Practicing graphing techniques
• Solving various algebraic equations, including absolute value equations

Remember, repetition and practice are key. The more you practice, the better prepared you will be for calculus and beyond.