Problem 50 - A Graphical Approach to College Algebra

Question

Solve each equation or inequality. $$5|x+3|-2=18$$

Step-by-Step Solution

Answer

The solutions are \(x = 1\) and \(x = -7\).

Step 1: Isolate the Absolute Value

The given equation is \(5|x+3|-2=18\). First, let's isolate the absolute value expression by adding 2 to both sides of the equation: \[5|x+3| = 20\]

Step 2: Solve for the Absolute Value

Divide both sides of the equation by 5 to solve for the absolute value:\[|x+3| = 4\]

Step 3: Set Up Two Equations

The absolute value equation \(|x+3|=4\) means that the expression inside the absolute value can be 4 or -4. So, we set up two separate equations: \(x+3 = 4\) and \(x+3 = -4\)

Step 4: Solve Each Equation

First, solve \(x+3=4\):Subtract 3 from both sides: \[x = 1\]Next, solve \(x+3=-4\):Subtract 3 from both sides:\[x = -7\]

Step 5: Verify Solutions

Substitute \(x=1\) back into the original equation and calculate:\[5|1+3|-2 = 5(4) -2 = 20 - 2 = 18\], which matches the right-hand side.Substitute \(x=-7\) back into the original equation and calculate:\[5|-7+3|-2 = 5|-4|-2 = 20 - 2 = 18\], which also matches.Both solutions satisfy the original equation.

Key Concepts

Isolating Absolute Value

When solving absolute value equations, it's crucial to start by isolating the absolute value expression. Think of it as peeling away the layers of an onion until you're left with the core, which in this case is the absolute value part. In our example, the equation given was \(5|x+3|-2=18\).

The goal in this first step is to get rid of any additional constants or coefficients attached to the absolute value. Here's how we do it:

• First, add 2 to both sides: \(5|x+3| = 20\). This neutralizes the \(-2\) on the left.
• Next, divide both sides by 5 to completely isolate \(|x+3|\): \(|x+3| = 4\). Now the absolute value is all alone on one side of the equation, making it easier to handle.

Isolating the absolute value simplifies the problem. After this step, the challenge becomes dealing with the absolute value expression itself.

Setting Up Equations

After isolating the absolute value, the next step is setting up equations. This step requires understanding what an absolute value means. The absolute value of a number represents its distance from zero on the number line. Hence, an expression like \(|x+3| = 4\) guides us to two scenarios:

• The expression \(x+3\) could be equal to 4, making the equation \(x+3=4\).
• Or the expression \(x+3\) might be equal to \(-4\), making the equation \(x+3=-4\).

Notice this step involves splitting one equation into two. It acknowledges that there are two potential solutions when it comes to absolute values – positive and negative distances from zero. After establishing these equations, the path forward is simply to solve each one independently.

Verifying Solutions

Now that you've set up and solved your equations, the final piece of the puzzle is verifying solutions. This critical step ensures your solutions are correct. Begin by substituting each potential solution back into the original equation to check if the left side equals the right side.

For the first solution \(x=1\):

• Plug it into the original equation: \(5|1+3|-2\).
• Simplify: \(5 \times 4 - 2 = 20 - 2 = 18\), which equals the original right-hand side, confirming \(x=1\) as a valid solution.

For the second solution \(x=-7\):

• Insert into the original equation: \(5|-7+3|-2\).
• This becomes: \(5 \times |-4| - 2 = 20 - 2 = 18\), again equaling the right-hand side, establishing \(x=-7\) as a valid solution.

Verification prevents errors and confirms both solutions truly satisfy the equation, showcasing that the problem-solving process is thorough and accurate.