Problem 35 - College Algebra

Question

For the following exercises, solve the equation involving absolute value. $$ |x+5|=0 $$

Step-by-Step Solution

Answer

The solution is \(x = -5\).

Step 1: Understand the Absolute Value Equation

Absolute value measures the distance a number is from zero on the number line, meaning it is always non-negative. The equation \(|x+5| = 0\) implies that the expression inside the absolute value, \(x+5\), must be exactly equal to zero because the only way an absolute value equals zero is if the inside expression itself is zero.

Step 2: Solve the Equation

Set the expression inside the absolute value equal to zero: \(x+5 = 0\). To solve for \(x\), we need to isolate \(x\) by performing inverse operations.

Step 3: Isolate x

Subtract 5 from both sides of the equation: \(x+5-5 = 0-5\) simplifies to \(x = -5\). Now, \(x\) is isolated, giving the solution to the equation.

Step 4: Verify the Solution

To ensure the solution is correct, substitute \(x = -5\) back into the original equation. The substitution gives \(|(-5)+5| = |0|\), which is \(0\). Since the original equation states \(|x+5| = 0\), and the result is \(0\), the solution \(x = -5\) is verified.

Key Concepts

Solving Absolute Value Equations

Solving equations with absolute values involves understanding what absolute value means. The absolute value of a number represents its distance from zero on the number line.

• This distance is always non-negative, so when dealing with equations like \(|x+5| = 0\), the expression inside the absolute value must equal zero.
• In this case, \(x+5 = 0\) because the only time the absolute value of something is zero is when that something itself is zero.

Solving these equations generally consists of removing the absolute value by directly setting the expression equal to zero. From that point, the equation can be treated as a simple algebraic expression where you aim to solve for the unknown variable.

Understanding and Using Inverse Operations

Inverse operations are fundamental tools in solving equations, particularly when you want to isolate a variable. They essentially "undo" mathematical operations.

• In the equation \(x+5 = 0\), the original operation is adding 5 to \(x\).
• The inverse operation of addition is subtraction, meaning you should subtract 5 from both sides to isolate \(x\).

This gives us \(x = -5\), which is the solution to our equation. Using inverse operations efficiently allows you to systematically work through any linear equation to find the unknown variable.

Verification of Solutions

Verifying solutions is a critical step to ensure the answer is correct. Once you solve an equation, you should always plug the solution back into the original equation.

• For the equation \(|x+5| = 0\), substitute \(x = -5\) back into the left-hand side of the equation.
• This gives \(|(-5) + 5|\) which simplifies to \(|0|\), and since the absolute value of zero is zero, it matches the original equation \(|x+5| = 0\).

By verifying, you're confirming that your calculations were correct and the solution fits seamlessly into the equation without any discrepancies. It's a reassurance that you've solved the problem accurately.