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Chapter 1: Problem 19

Find the real solutions, if any, of each equation. $$ \frac{8}{7}|x|=3 $$

Short Answer

Expert verified
The solutions are \( x = \frac{21}{8} \) and \( x = -\frac{21}{8} \).

Step by step solution

01

Isolate the Absolute Value Expression

Start by isolating the absolute value expression on one side of the equation. The given equation is \ \( \frac{8}{7}|x|=3 \). Multiply both sides by \( \frac{7}{8} \) to get \( |x| \). \ \( |x| = 3 \cdot \frac{7}{8} \) \ \( |x| = \frac{21}{8} \)
02

Split the Equation into Two Cases

Since \( |x| = \frac{21}{8} \), we need to consider both the positive and negative cases of the equality. \ Case 1: \( x = \frac{21}{8} \) \ Case 2: \( x = -\frac{21}{8} \)
03

Write the Final Solution

Combine the solutions from both cases: \ Thus, \( x = \frac{21}{8} \) or \( x = -\frac{21}{8} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value
The absolute value of a number, represented as \(|x|\), is its distance from zero on the number line, regardless of direction. It means \(|x|\) outputs only non-negative numbers. For example:
  • \(|-5| = 5\)
  • \(|3| = 3\)
This property is crucial in solving equations involving absolute values because you need to consider both the positive and negative scenarios for the variable inside the absolute value brackets.
real solutions
Real solutions are solutions to an equation that are real numbers. In our problem, the equation \[ \frac 87 |x| = 3 \] has real numbers as solutions. To find these solutions, we need to manipulate the equation to isolate the variable. Once isolated, we consider both the positive and negative values, as real solutions can encompass both.
algebraic manipulation
Algebraic manipulation refers to the steps taken to isolate variables in an equation. Let's see this in our example:
1. The original problem is \[ \frac 87 |x| = 3 \]
2. We multiply both sides by the reciprocal of \frac{8}{7}, which is \frac{7}{8}.
3. This leaves us with \[ |x| = 3 \frac{7}{8} \]
4. Simplifying, \[ |x| = \frac{21}{8} \]
We now have isolated \(|x|\) and are ready for the next step.
case analysis
Case analysis involves breaking down a problem into separate scenarios and solving each one. Because \(|x|\) represents both positive and negative distances from zero, we need to solve for both \[ x \] and \[ -x \].
1. We found \[ |x| = \frac{21}{8} \]
2. Consider the positive case: \[ x = \frac{21}{8} \]
3. Consider the negative case: \[ x = -\frac{21}{8} \]
By combining these, we get our full solution set: \[ x = \frac{21}{8} \] or \[ x = -\frac{21}{8} \]. Always check your answers to ensure they fit the original equation.

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Most popular questions from this chapter

A Biathlon Suppose that you have entered an 87 -mile biathlon that consists of a run and a bicycle race. During your run, your average speed is 6 miles per hour, and during your bicycle race, your average speed is 25 miles per hour. You finish the race in 5 hours. What is the distance of the run? What is the distance of the bicycle race?

Find the real solutions, if any, of each equation. $$ \sqrt[4]{4-3 x^{2}}=x $$

Arithmetic Mean If \(a

In the 2016 Olympics, Usain Bolt of Jamaica won the gold medal in the 100 -meter race with a time of 9.81 seconds. In the 1896 Olympics, Thomas Burke of the United States won the gold medal in the 100-meter race in 12.0 seconds. If they ran in the same race, repeating their respective times, by how many meters would Bolt beat Burke?

Find \(k\) so that the equation \(x^{2}-k x+4=0\) has a repeated real solution.
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