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Chapter 1: Problem 3

Solve for \(\mathrm{x}\) when \(\quad(3 / \mathrm{x})<5 . \quad \mathrm{x} \neq 0\)

Short Answer

Expert verified
The solution for x in the inequality \(\frac{3}{x}<5\) with \(x\neq 0\) is \(x \in \left(\frac{3}{5}, +\infty\right)\).

Step by step solution

01

Rewrite the inequality

In order to manipulate the inequality, rewrite it with the variable x in the numerator: \[\frac{3}{x} < 5\]
02

Eliminate the fraction

To get rid of the fraction, multiply both sides of the inequality by x. Since x 鈮� 0, the inequality sign doesn't change. The inequality becomes: \[3 < 5x\]
03

Isolate x

Divide both sides of the inequality by 5 to isolate x: \[\frac{3}{5} < x\]
04

Write the solution

Finally, we can express the solution for x in interval notation: \(x \in \left(\frac{3}{5}, +\infty\right)\) So, x can be any value from the interval \(\left(\frac{3}{5}, +\infty\right)\), but x cannot equal 0.

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Key Concepts

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

Manipulating Inequalities
When solving inequalities, sometimes you need to manipulate the expression to simplify or rearrange it. This process is essential for understanding and solving the inequality. In the given exercise, we start with the inequality \( \frac{3}{x} < 5 \), where our goal is to solve for \( x \).
To manipulate an inequality effectively, you might need to:
  • Switch between different forms of the expression, such as converting fractions to a simpler form.
  • Apply algebraic operations, like adding, subtracting, multiplying, or dividing terms on both sides of the inequality.
  • Maintain the inequality relationship. Be mindful that the inequality's direction changes if you multiply or divide both sides by a negative number.
In this exercise, by eventually aiming to have \( x \) out of the denominator, we streamline our path to isolating \( x \) in subsequent steps.
Fraction Elimination
Eliminating fractions is a crucial part when solving inequalities involving fractions. It makes the inequality easier to handle by removing the fraction.
For the inequality \( \frac{3}{x} < 5 \), you multiply both sides by \( x \) to eliminate the fraction. This works under the condition that \( x eq 0 \) given in the problem.
  • Ensure that you don't alter the inequality when dealing with positive \( x \).
  • If \( x \) was negative (hypothetically), the inequality would flip - crucial in other scenarios.
This step simplifies the inequality to \( 3 < 5x \), paving the way to isolate \( x \) more straightforwardly.
Interval Notation
Once you find the solution to an inequality, expressing it in interval notation gives a clear view of all possible solutions. Interval notation is a mathematical way of describing a set of numbers along a number line.
To express \( x > \frac{3}{5} \):
  • The lower bound is \( \frac{3}{5} \) (exclusive), indicating that \( x \) starts from \( \frac{3}{5} \) but does not include it.
  • The upper bound is \( +\infty \), meaning \( x \) can continue indefinitely.
In interval notation, it is expressed as \( \left(\frac{3}{5}, +\infty\right) \). This tells us \( x \) can take any value in this range, giving a compact and complete representation of all possible solutions.
Variable Isolation
Isolating a variable means manipulating an equation or inequality so that the variable appears alone on one side. This step is often the final part of solving an inequality to find the solution set.
For \( 3 < 5x \), you divide both sides by 5 to free \( x \) from other coefficients:
  • Divide: \( \frac{3}{5} < x \), simplifies to isolate the variable.
  • When dividing, remember division by a positive number keeps the inequality sign unchanged.
This results in \( x > \frac{3}{5} \), which is simpler and directly usable for expressing in interval notation.

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Most popular questions from this chapter

Solve the inequality: \(2 \mathrm{x}^{2}+5 \mathrm{x}<12\)

Graph the following inequalities: \((x+1)^{2}+y^{2}<1, \quad\) and \(\quad(x+1)^{2}+y^{2}>1\)

Solve for \(\mathrm{x}(\mathrm{x} \neq-2)\) when \([(2 \mathrm{x}-3) /(\mathrm{x}+2)]<(1 / 3)\)

Draw the graph of the equation: \(\mathrm{x}+\mathrm{y}=5\) and describe the inequalities that apply to the regions of the plane which this line separates.

Solve the inequality: \(\mathrm{x}^{2}>\mathrm{x}\).
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