Problem 42 - Intermediate Algebra

Question

Solve each equation. See Example 3. $$ 8\left|\frac{2 x}{3}+10\right|=0 $$

Step-by-Step Solution

Answer

The solution is \( x = -15 \).

Step 1: Understand the Absolute Value Property

The equation involves an absolute value. The absolute value of a number is always non-negative. Therefore, for an equation of the form \( |A| = k \), the equation is only solvable when \( k \geq 0 \). Here, the right hand side is 0, which is the minimum value for absolute values.

Step 2: Set Up Equation for Zero Condition

Since the absolute value part must equal zero (the only scenario where the absolute value is zero is when what's inside the absolute value also equals zero), set the inside of the absolute value to zero: \( \frac{2x}{3} + 10 = 0 \).

Step 3: Solve the Equation Inside the Absolute Value

Solve the equation \( \frac{2x}{3} + 10 = 0 \) for \( x \). Start by isolating \( x \): subtract 10 from both sides to get \( \frac{2x}{3} = -10 \).

Step 4: Clear the Fraction and Solve for x

Multiply both sides of the equation by 3 to eliminate the fraction: \( 2x = -30 \). Then, divide both sides by 2 to solve for \( x \): \( x = -15 \).

Step 5: Verify the Solution

Substitute \( x = -15 \) back into the original expression to verify: \( \frac{2(-15)}{3} + 10 = \frac{-30}{3} + 10 = -10 + 10 = 0 \). The equality holds, confirming \( x = -15 \) is the correct solution.

Key Concepts

Absolute Value Property

Absolute value represents the distance of a number from zero on the number line. It is always non-negative, because distance cannot be negative. This means that for an expression inside an absolute value to equal zero, the value itself must be zero.
In the exercise, we have the equation \( 8\left|\frac{2x}{3}+10\right|=0 \). For the equation to hold true, the absolute value expression \( \left|\frac{2x}{3}+10\right| \) must equal zero. This is because multiplying any positive number (in this case, 8) by a number cannot give zero unless the number itself is zero.
So, to solve the equation, we need to ensure that what is inside the absolute value bracket equals zero.

• The principle is: \(|A| = 0\) implies \(A = 0\).
• Recognizing when the inside of the absolute value equals zero is crucial to solving these types of equations.

Isolation of Variable

The next key step involves isolating the variable present in the equation. We start from the evaluated condition \( \frac{2x}{3}+10 = 0 \).
To simplify, follow these steps:

• First, subtract 10 from both sides to remove the constant term. This gives us \( \frac{2x}{3} = -10 \).
• It's helpful to work with fewer fractions, so multiply through by 3 to clear the fraction: \( 2x = -30 \).
• Finally, divide by 2 to isolate \( x \) on one side: getting \( x = -15 \).

Each step here involves basic algebraic manipulation, such as adding, subtracting, multiplying, or dividing to balance both sides of the equation. It's important to apply operations to both sides equally to maintain equality. Working step-by-step helps simplify finding \( x \).

Verification of Solution

Verification ensures that the solution satisfies the original equation. Once we have a potential solution, \( x = -15 \), it's essential to plug it back into the initial equation to confirm.
Substituting \( x = -15 \) into the original equation part involving the absolute value:

• Calculate: \( \frac{2(-15)}{3} + 10 \).
• Simplify: \( \frac{-30}{3} + 10 = -10 + 10 \).
• Verify: Result is 0, which matches the equation \( 8\left|\frac{2x}{3}+10\right|=0 \).

This confirms that the calculation is correct and verifies \( x = -15 \) as the solution. Verifying by substitution is a reliable method to double-check correctness and to ensure that no errors occurred in calculation. Always remember that verifying the solution is the final step to solve all kinds of equations, ensuring they actually meet the conditions described in the original problem.