Problem 47 - Algebra for College Students

Question

Solve each equation and inequality. \(|x+2|-6=-2\)

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -6\).

Step 1: Isolate the Absolute Value

Start by adding 6 to both sides of the equation to isolate the absolute value:\(|x+2|-6+6 = -2+6\)This simplifies to:\(|x+2| = 4\)

Step 2: Set Up the Two Cases for the Absolute Value

Absolute value equations like \(|x+2| = 4\) result in two possible equations. Set up the two cases:Case 1: \(x+2 = 4\)Case 2: \(x+2 = -4\)

Step 3: Solve the First Equation

Solve the first case by subtracting 2 from both sides:\(x+2 = 4\)\(x = 4 - 2\)\(x = 2\)

Step 4: Solve the Second Equation

Solve the second case by subtracting 2 from both sides:\(x+2 = -4\)\(x = -4 - 2\)\(x = -6\)

Step 5: Verify the Solutions

Plug both values of \(x\) back into the original equation to verify they satisfy it:For \(x = 2\), check:\(|2+2|-6 = |4|-6 = 4-6 = -2\)The equation holds.For \(x = -6\), check:\(|-6+2|-6 = |-4|-6 = 4-6 = -2\)The equation holds.

Key Concepts

Solving Equations

When solving equations, especially those with absolute values, it's important to follow a structured approach. Absolute value equations often have two potential solutions. Why? Because absolute values represent distance from zero on the number line, making two cases possible. Here's what you generally do:

• First, isolate the absolute value on one side of the equation. This sets the stage for separating the equation into its two potential cases.
• After isolating, split the absolute value equation into these two linear equations: one where the inside of the absolute value is equal to the given number, and one where it is equal to the negative of that number.

This approach helps us account for all possible solutions and ensures no valid answer is overlooked.

Algebra Concepts

Understanding the concept of absolute value is crucial in algebra. An absolute value generally refers to the magnitude of a number or how far it is from zero, regardless of direction. In equations, it suggests there could be two scenarios for each magnitude, a positive and a negative.Algebra is a language of symbols and rules. It helps us translate real-world scenarios into mathematical expressions. To navigate the solutions for absolute value equations like \(|x+2| = 4\), you need to break down the equation into manageable parts. The symbolic representation boils down to setting up two different situations:

• One where the inside expression equals the positive case.
• Another where the expression equals the negative case.

Recognizing these underlying algebraic principles helps in methodically tackling such problems.

Student Practice Problems

Practicing with absolute value equations can strengthen your algebra skills and build confidence. Here are some tips to make practice more effective:

• Start with simpler equations to ensure you understand isolating the absolute value and setting up two cases.
• Gradually increase complexity by adding more variables or terms.
• Always check your answers by substituting potential solutions back into the original equation. This verifies your solutions both conceptually and computationally.

By consistently applying these strategies, students can enhance their problem-solving abilities and become more adept at figuring out absolute value problems. With practice, even the seemingly complex problems can become intuitive and easy to solve.