Problem 11 - Algebra and Trigonometry

Question

\(5-22=\) Solve the equation. $$ |x+4|=0.5 $$

Step-by-Step Solution

Answer

The solutions to the equation are \\(x = -3.5\\) and \\(x = -4.5\\).

Step 1: Understanding Absolute Value

The absolute value of a number refers to its distance from 0 on the number line, regardless of direction. For any positive number, changes within the absolute value expression can result in two possible equations because both positive and negative numbers have the same absolute value.

Step 2: Set Up Equations from Absolute Value Expression

To solve \(|x+4|=0.5\), recognize that this means both \((x+4) = 0.5\) and \((x+4) = -0.5\) could be true. These two possibilities need to be solved separately.

Step 3: Solving First Equation

Start with the equation \((x+4) = 0.5\). Subtract 4 from both sides to isolate \(x\); result is \(x = 0.5 - 4\). Perform the subtraction to find \(x = -3.5\).

Step 4: Solving Second Equation

Now solve the second equation \((x+4) = -0.5\). Subtract 4 from both sides to isolate \(x\); the result is \(x = -0.5 - 4\). Perform the subtraction to find \(x = -4.5\).

Step 5: Verify Solutions

To confirm both solutions are correct, substitute \(x = -3.5\) and \(x = -4.5\) back into the original absolute value equation. Both should evaluate to 0.5, thus confirming the solutions are valid.

Key Concepts

Distance on Number Line

When thinking about absolute value, it's crucial to visualize the number line. The absolute value of a number refers to its distance from zero. Imagine standing at zero and stepping either to the right or left. Each step is a positive unit, meaning absolute value is always non-negative.
This can be particularly helpful when dealing with equations containing absolute values. For example, \(|x+4| = 0.5\) suggests two distances on the number line.
We understand this as either \(x+4\) being 0.5 steps to the right of zero or \(x+4\) being 0.5 steps to the left, leading to two separate equations: \(x+4 = 0.5\) and \(x+4 = -0.5\).

• This duality is key because the absolute value doesn't differentiate between direction, only distance.
• Being comfortable with visualizing numbers on a line can enhance your understanding of these concepts.

Solving Equations

Solving absolute value equations involves unfolding the expression into two linear equations. Once you recognize that \(|x+4|=0.5\) indicates two possible scenarios, you can tackle them separately.
For instance, if \(x+4=0.5\), solving it means isolating \(x\). Subtract 4 from both sides to find \(x=-3.5\).
Similarly, for \(x+4=-0.5\), perform the same operation (subtract 4) on both sides to find \(x=-4.5\).

• This method ensures all potential solutions from the absolute value are accounted for.
• Each step involves simple arithmetic manipulation, which helps in breaking down the problem.

Verification of Solutions

After you've solved the equations, it's essential to verify the solutions by substituting them back into the original equation. This step confirms the validity of your findings.
Take \(x=-3.5\) and plug it back into \(|x+4|=0.5\). Calculate \(|-3.5+4|=|0.5|=0.5\), which holds true.
Then, try \(x=-4.5\). Substitute to get \(|-4.5+4|=|-0.5|=0.5\), which again validates the solution.

• Verification ensures no potential errors occurred during computation.
• It's also a good practice to help you build confidence in solving equations.