Problem 17 - Algebra and Trigonometry

Question

\(5-22=\) Solve the equation. $$ 3|x+5|+6=15 $$

Step-by-Step Solution

Answer

The solution set is \(x = -2\) and \(x = -8\).

Step 1: Isolate the Absolute Value Expression

First, subtract 6 from both sides of the equation to begin isolating the absolute value expression. This yields:\[3|x+5| + 6 - 6 = 15 - 6\]which simplifies to:\[3|x+5| = 9\]

Step 2: Solve for the Absolute Value

Next, divide both sides by 3 to completely isolate the absolute value expression:\[\frac{3|x+5|}{3} = \frac{9}{3}\]which simplifies to:\[|x+5| = 3\]

Step 3: Solve for x in Positive Case

The absolute value equation can now be split into two separate cases. The first case is when the expression inside the absolute value is equal to 3:\[x+5 = 3\]Subtract 5 from both sides:\[x = 3 - 5\]\[x = -2\]

Step 4: Solve for x in Negative Case

The second case is when the expression inside the absolute value is equal to -3:\[x+5 = -3\]Subtract 5 from both sides:\[x = -3 - 5\]\[x = -8\]

Step 5: Conclusion

The solutions are the values of \(x\) obtained from both the cases:\[x = -2\] and \[x = -8\]. Therefore, the equation is satisfied for \(x = -2\) and \(x = -8\).

Key Concepts

Isolate the Absolute Value

When you're solving absolute value equations, the first step is to isolate the absolute value expression. This means you want to get the absolute value term by itself on one side of the equation. In this exercise, we start with the equation:
\[ 3|x+5| + 6 = 15 \]
Our goal is to first get \( |x+5| \) alone on one side.

• Subtract 6 from both sides to eliminate the constant term outside the absolute value. This gives us: \( 3|x+5| = 9 \).
• Next, divide both sides by 3, which simplifies the equation to: \( |x+5| = 3 \). Now the absolute value is completely isolated.

By isolating the absolute value, we've prepared the equation for the next critical steps.

Solve for x

With the absolute value expression now isolated as \( |x+5| = 3 \), the next task is to "solve for x." The absolute value operation creates two distinct scenarios because it represents the distance from zero, which can be either positive or negative.

• First, consider the situation where \( x+5 \) is equal to the positive value of 3. This means solving: \( x+5 = 3 \). By subtracting 5 from both sides, we find: \( x = -2 \).
• Next, handle the scenario where \( x+5 \) equals the negative value, which is \(-3\). Set up the equation: \( x+5 = -3 \). Again, subtract 5 from each side to reveal: \( x = -8 \).

These calculations show how you can determine the possible values of \( x \) that satisfy the original equation. Solving for \( x \) in this way gives you a complete solution.

Positive and Negative Cases

When dealing with absolute value equations, understanding that each absolute value equation has two potential solutions due to its nature is essential. Once the absolute value is isolated, it shows distances on a number line that need to be addressed by two scenarios: the positive and the negative cases.

• The positive case assumes the expression inside the absolute value is equal to the positive side. For \( |x+5| = 3 \), this gives \( x+5 = 3 \), leading to \( x = -2 \).
• The negative case deals with when the expression is equal to the negative side. Here, it’s \( x+5 = -3 \), resulting in \( x = -8 \).

Both solutions need to be considered and checked back with the original equation. This ensures both values make the equation true and completes the exercise by considering all possibilities given by the absolute value.