Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 0: Problem 17
Find the solution set, graph this set on the real line, and express this set in interval notation. $$ \frac{3}{1-x} \leq 2 $$
Short Answer
Expert verified
The solution set is \((-\infty, -\frac{1}{2}]\).
Step by step solution
01
Set Up the Inequality
We start with the given inequality \( \frac{3}{1-x} \leq 2 \). Our goal is to solve for \( x \). To do this, we need to clear the fraction first.
02
Clear the Fraction
Multiply both sides of the inequality by \( 1-x \) to eliminate the fraction. However, since \( 1-x \) may be negative, we must account for this when multiplying. Let's first consider when \( 1-x > 0 \), which implies \( x < 1 \): \( 3 \leq 2(1-x) \).
03
Solve the Inequality for x < 1
Distribute the right-hand side: \( 3 \leq 2 - 2x \).Now solve for \( x \) by subtracting 2 from both sides: \( 1 \leq -2x \).Divide by \( -2 \) and remember to flip the inequality sign: \( x \leq -\frac{1}{2} \).
04
Consider the Case When x = 1
If \( x = 1 \), then the expression \( \frac{3}{1-x} \) becomes undefined, which means \( x = 1 \) cannot be part of the solution set.
05
Write the Solution Set
Combining results, the solution set for the inequality is \( x \leq -\frac{1}{2} \). The restriction \( x < 1 \) doesn't change the solution since \( x \leq -\frac{1}{2} \) is already smaller than 1.
06
Express in Interval Notation
Expressing the solution \( x \leq -\frac{1}{2} \) in interval notation gives \( (-\infty, -\frac{1}{2}] \).
07
Graph the Solution on the Real Line
On a number line, shade all values to the left of \(-\frac{1}{2}\) and include \(-\frac{1}{2}\) with a filled dot, indicating that it is part of the solution set.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Fractions
When solving inequalities involving fractions, such as \( \frac{3}{1-x} \leq 2 \), the primary goal is to clear the fraction to simplify the equation. This is achieved by eliminating the denominator through multiplication. However, caution is warranted because multiplication or division by a negative number necessitates flipping the inequality sign.
- Identify the denominator: In \( \frac{3}{1-x} \), 1-x is the denominator.
- Clear the fraction: Multiply through by \( 1-x \), ensuring to consider the sign of this term as it affects the inequality.
- Handle cases: Determine if \( 1-x \) is positive or negative by solving \( 1-x=0 \) to find boundary conditions.
Interval Notation
Interval notation offers a streamlined method to express solution sets of inequalities. This approach captures the set of all numbers satisfying the inequality in a concise manner. Consider the inequality result \( x \leq -\frac{1}{2} \).
- Use interval brackets: Square brackets \([ \text{ ]} \) are used when the number is included (\( \leq, \geq \)), whereas parentheses \(( \text{ )} \) signify exclusion.
- Identify extremes: For \( x \leq -\frac{1}{2} \), the solution is all numbers from negative infinity up to and including \(-\frac{1}{2} \).
- Express the range: This is written as \( (-\infty, -\frac{1}{2}] \).
Real Number Line
Graphing on the real number line helps visualize solution sets in a clear manner. The real number line extends infinitely in both directions, going left for negative numbers and right for positive numbers. Visualizing the solution, \( x \leq -\frac{1}{2} \), involves a straightforward shading process.
- Identify the boundary: Locate \(-\frac{1}{2} \) on the number line.
- Shade the solution set: Shade all numbers to the left of \(-\frac{1}{2} \), representing numbers \( x \leq -\frac{1}{2} \), while using a filled dot at \(-\frac{1}{2} \) to denote inclusion in the set.
- Understand infinite continuation: The shading extends to the left indefinitely, in line with \( (-\infty, -\frac{1}{2}] \).
