1-4 Study Guide and Intervention: Solving Absolute Value Equations

Introduction to Absolute Value

Absolute value is a fundamental concept in mathematics that represents the distance of a number from zero on a number line, regardless of direction. In simpler terms, the absolute value of any number is always positive or zero. Mathematically, the absolute value of a number $x$ is denoted as $|x|$, and it’s defined as:

• $|x|=x$ if $x\ge 0$
• $|x|=-x$ if $x<0$

For example, the absolute value of $|5|$ is 5, and the absolute value of $|-5|$ is also 5.

In this study guide, we’ll dive deeper into solving absolute value equations, an essential skill in algebra.

Absolute Value Equations: The Basics

An absolute value equation is an equation that contains an absolute value expression. A typical form of such an equation is:

$$|A|=B$$

Where $A$ is an algebraic expression and $B$ is a positive real number. To solve this equation, we must understand that $A$ could equal $B$ or $-B$. This is because the absolute value strips away the sign, meaning both positive and negative values of $A$ will satisfy the equation.

For instance, in the equation $|x|=4$, the solutions are:

$$x=4\text{or}x=-4$$

This is the key idea behind solving absolute value equations.

Key Steps to Solve Absolute Value Equations

Follow these steps to solve an absolute value equation:

1. Isolate the absolute value expression: Before solving, ensure the absolute value expression is by itself on one side of the equation.
2. Set up two cases: Since $|A|=B$ implies that $A=B$ or $A=-B$, you must create two separate equations.
3. Solve each equation: Solve both equations from the previous step.
4. Check for extraneous solutions: When solving absolute value equations, especially those that involve variables in more complex expressions, always verify that the solutions are valid by substituting them back into the original equation.

Example 1: Simple Absolute Value Equation

Let’s solve a basic equation:

$$|x-3|=7$$

Step 1: Isolate the absolute value expression
In this case, the absolute value expression $|x-3|$ is already isolated.

Step 2: Set up two cases
We break this into two equations:

1. $x-3=7$
2. $x-3=-7$

Step 3: Solve each equation

For $x-3=7$:

$$x=7+3=10$$

For $x-3=-7$:

$$x=-7+3=-4$$

Step 4: Check for extraneous solutions
Both solutions $x=10$ and $x=-4$ satisfy the original equation, so they are valid solutions.

Thus, the solution set is:

$$x=10\text{or}x=-4$$

Example 2: Absolute Value Equation with Variables

Let’s solve a slightly more complex equation:

$$2|3x-5|+4=14$$

Step 1: Isolate the absolute value expression
First, subtract 4 from both sides:

$$2|3x-5|=10$$

Next, divide both sides by 2:

$$|3x-5|=5$$

Step 2: Set up two cases
Now, break the equation into two cases:

1. $3x-5=5$
2. $3x-5=-5$

Step 3: Solve each equation

For $3x-5=5$:

3x=5+5=10⇒x=1033x = 5 + 5 = 10 \quad \Rightarrow \quad x = \frac{10}{3}3x=5+5=10⇒x=310​

For $3x-5=-5$:

$$3x=-5+5=0\Rightarrow x=0$$

Step 4: Check for extraneous solutions
Both solutions x=103x = \frac{10}{3}x=310​ and $x=0$ are valid because they satisfy the original equation. Thus, the solution set is:

x=103orx=0x = \frac{10}{3} \quad \text{or} \quad x = 0x=310​orx=0

Example 3: No Solution Scenario

Sometimes, absolute value equations can have no solution. For example:

$$|x+2|=-3$$

Since the absolute value of any expression is always non-negative, it is impossible for $|x+2|$ to equal a negative number. Therefore, this equation has no solution.

Common Mistakes to Avoid

• Forgetting to check for extraneous solutions: Not every solution derived from an absolute value equation is valid. Always plug your solutions back into the original equation to verify.
• Ignoring negative solutions: Absolute value equations often have two solutions. Be sure to account for both the positive and negative cases.

Applications of Absolute Value Equations

Absolute value equations appear in various real-world contexts, particularly when modeling scenarios involving distance or magnitude. For example:

• Geometry: The absolute value is used to measure the distance between points on a number line.
• Physics: In physics, absolute values are useful for representing magnitudes such as velocity, force, or displacement without regard to direction.
• Economics: In economics, absolute values are often used to represent deviations or variances from target values or means, regardless of the direction of the deviation.

Conclusion

Solving absolute value equations is a vital skill in algebra that requires careful attention to isolating the absolute value expression, setting up two cases, solving both, and checking for extraneous solutions. These equations can sometimes have no solution, so it’s important to interpret the results correctly. By mastering these steps, you’ll be well-equipped to handle both basic and more complex absolute value problems.