Problem 27 - College Algebra with Modeling and Visualization

Question

Solve the absolute value equation. $$|7-16 x|=0$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{7}{16}\).

Step 1: Understanding Absolute Value Equation

The absolute value equation given is \(|7 - 16x| = 0\). The absolute value of an expression is zero when the expression itself is zero. This means we have to set the expression inside the absolute value equal to zero: \(7 - 16x = 0\).

Step 2: Solving for x

To solve for \(x\), we need to isolate \(x\) on one side of the equation. Start by moving the constant term to the other side: \(7 = 16x\). Then, divide both sides by 16 to solve for \(x\): \(x = \frac{7}{16}\).

Step 3: Verification

To verify the solution, substitute \(x = \frac{7}{16}\) back into the original absolute value equation: \(|7 - 16 \times \frac{7}{16}| = 0\). Calculate inside the absolute value: \(7 - 7 = 0\). The absolute value of 0 is 0, confirming our solution is correct.

Key Concepts

Solving Equations

When solving equations, it's crucial to understand what you're working with. Here, we are tackling an absolute value equation. The equation provided is \(|7 - 16x| = 0\). The goal is to find the value of \(x\) that makes this equation true.

In absolute value equations, remember that the expression inside the absolute value will be zero since the absolute value of zero is zero. Therefore, we start by setting the equation inside the absolute value bars to zero: \(7 - 16x = 0\). This step simplifies our work to a basic algebraic equation.

Next, manipulate the equation to isolate \(x\). Begin by subtracting 7 from both sides, resulting in \(16x = 7\). Then, divide both sides by 16 to solve for \(x\):

• Subtract 7: \(-7 = -16x\)
• Switch sides: \(16x = 7\)
• Division step: \(x = \frac{7}{16}\)

As a result, we've found our solution by breaking down the problem into simpler operations.

Verification of Solution

Verifying an equation solution means checking that your answer satisfies the original equation. This ensures that the solution is indeed correct. For our equation, we found \(x = \frac{7}{16}\) as the solution. Now, we will substitute this value of \(x\) back into the original absolute value equation to verify it.

Plug the solution back into the equation: \(|7 - 16 \times \frac{7}{16}| = 0\). Now perform the necessary simplification and calculation steps to make sure the equation holds true:

• Calculate the expression: \(7 - (16 \times \frac{7}{16}) = 7 - 7 = 0\)
• Apply absolute value: \(|0| = 0\)

Since substituting \(x = \frac{7}{16}\) gives us \(0\), which matches the right-hand side of the equation, our solution is confirmed to be correct.

Algebraic Expressions

Algebraic expressions form the building blocks for many types of equations, including those involving absolute values. In our exercise, the expression \(7 - 16x\) is at the core of the equation.

When dealing with algebra, expressions are combinations of numbers, variables, and operations. They represent "quantities" and can be manipulated following algebraic principles. Here we focus on solving expressions set inside an absolute value equation.

Manipulating expressions involves different operations like addition, subtraction, multiplication, or even division, just as we did to isolate \(x\) in our solution:

• Recognize similar terms and move constants: \(7\)
• Switch terms across the equality: \(16x\)
• Solve through division or multiplication to consolidate terms: from \(16x = 7\) to \(x = \frac{7}{16}\)

The essence of algebraic work is transforming expressions to uncover the values of variables. This showcases the utility and elegance of algebra in solving real-world mathematical problems.