Problem 26

Question

Solve each equation. $\left|\frac{x}{7}\right|=2$

Step-by-Step Solution

Verified

Answer

The solutions are $ x = 14 $ and $ x = -14 $.

1Step 1: Understand the Absolute Value Equation

The absolute value equation given is $ \left| \frac{x}{7} \right| = 2 $. This means that the expression inside the absolute value, $ \frac{x}{7} $, can be either 2 or -2 for the equation to hold true.

2Step 2: Set Up the Two Possible Equations

We need to solve for $ x $ in both scenarios: $ \frac{x}{7} = 2 $ and $ \frac{x}{7} = -2 $.

3Step 3: Solve the First Equation

For $ \frac{x}{7} = 2 $, multiply both sides by 7 to isolate $ x $: \[ x = 2 \times 7 = 14 \].

4Step 4: Solve the Second Equation

For $ \frac{x}{7} = -2 $, multiply both sides by 7 to isolate $ x $: \[ x = -2 \times 7 = -14 \].

5Step 5: Conclusion

The solutions to the equation $ \left| \frac{x}{7} \right| = 2 $ are $ x = 14 $ and $ x = -14 $.

Key Concepts

Solving EquationsAlgebraic ExpressionsStep-by-Step Solutions

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes an equation true. When dealing with absolute value equations like $ |\frac{x}{7}| = 2 $, we need to consider what the absolute value function represents. Absolute value measures the distance a number is from zero on the number line, regardless of direction. Therefore, both positive and negative values can emerge as solutions.

To solve $ |\frac{x}{7}| = 2 $, we express the equation as two distinct cases. The first case is straightforward, where the expression inside the absolute value $ \frac{x}{7} $ equals 2. The second case occurs when $ \frac{x}{7} $ equals -2. This bifurcation is necessary because of the nature of absolute values. Each case is then individually solved, leading to two possible solutions.

Algebraic Expressions

An algebraic expression involves numbers, variables, and arithmetic operations. In our exercise, the algebraic expression is $ \frac{x}{7} $. To work with an expression wrapped in absolute value, we must distill it into simpler forms.

Algebraic expressions allow us to codify mathematical statements and solve for unknowns. Simplifying expressions involves rearranging and manipulating terms to isolate the variable, like $ x $. For an absolute value equation, breaking it into possible distinct algebraic expressions ($ \frac{x}{7} = 2 $ and $ \frac{x}{7} = -2 $) helps in managing the inherent duality of absolute values.

Step-by-Step Solutions

Approaching mathematics with a step-by-step methodology paves the way for clarity and understanding. By dissecting complex problems into manageable parts, students can navigate through the solution process more effortlessly.

Here's how the problem is tackled:

Identify: Recognize the equation $ |\frac{x}{7}| = 2 $ as one involving absolute values.
Translate: Convert the single absolute value equation into two linear equations: $ \frac{x}{7} = 2 $ and $ \frac{x}{7} = -2 $.
Calculate: Solve each linear equation by performing inverse operations like multiplication to isolate $ x $.
Conclude: Combine solutions from both scenarios: $ x = 14 $ and $ x = -14 $.

This structured strategy is particularly beneficial for visual learners and those who prefer a logical progression to problem-solving.

Problem 26

Other exercises in this chapter

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Problem 26

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Problem 26

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