/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Express the solution set of the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 19

Express the solution set of the given inequality in interval notation and sketch its graph. $$ \frac{1}{3 x-2} \leq 4 $$

Short Answer

Expert verified
The solution set is \((-\infty, \frac{2}{3}) \cup [\frac{3}{4}, \infty)\).

Step by step solution

01

Write the Inequality

Start by writing the inequality: \( \frac{1}{3x-2} \leq 4 \). We need to find the values of \( x \) that satisfy this inequality.
02

Reorganize the Inequality

To isolate the fraction, we first subtract 4 from both sides: \( \frac{1}{3x-2} - 4 \leq 0 \). We can express the left side with a common denominator: \( \frac{1 - 4(3x-2)}{3x-2} \leq 0 \), which simplifies to: \( \frac{1 - 12x + 8}{3x-2} \leq 0 \).
03

Simplify the Inequality

Simplify the expression: \( \frac{9 - 12x}{3x-2} \leq 0 \). This will help us find the critical values where the expression equals zero or is undefined.
04

Identify Critical Points

Set the numerator to zero: \( 9 - 12x = 0 \). Solving for \( x \) gives \( x = \frac{3}{4} \). Set the denominator to zero: \( 3x - 2 = 0 \). Solving for \( x \) gives \( x = \frac{2}{3} \). These critical points divide the number line into intervals.
05

Test Intervals

Test the intervals formed by the critical points \( x = \frac{2}{3} \) and \( x = \frac{3}{4} \). Choose test values from the intervals: on the interval \( (-\infty, \frac{2}{3}) \), test \( x = 0 \); on \( (\frac{2}{3}, \frac{3}{4}) \), test \( x = \frac{5}{6} \); and on \( (\frac{3}{4}, \infty) \), test \( x = 1 \).
06

Evaluate Test Points

For each test point, check if \( \frac{9 - 12x}{3x-2} \leq 0 \):- For \( x = 0 \), \( \frac{9 - 12(0)}{3(0) - 2} = -\frac{9}{2} \) (Negative).- For \( x = \frac{5}{6} \), \( \frac{9 - 12(\frac{5}{6})}{3(\frac{5}{6}) - 2} > 0 \)(Positive).- For \( x = 1 \), \( \frac{9 - 12(1)}{3(1) - 2} < 0 \)(Negative).
07

Determine Interval of Solution

The inequality \( \frac{9 - 12x}{3x-2} \leq 0 \) holds for intervals where the expression is negative or zero. Thus, solution intervals are \( (-\infty, \frac{2}{3}) \cup (\frac{3}{4}, \infty) \). Since \( x = \frac{3}{4} \) makes the expression zero, include it.
08

Express Solution in Interval Notation

The solution set is \[ (-\infty, \frac{2}{3}) \cup [\frac{3}{4}, \infty) \].
09

Sketch the Graph

To sketch the solution on a number line: - Use an open circle at \( \frac{2}{3} \) (since it's not included in the solution set).- Use a closed circle at \( \frac{3}{4} \) (since it's included).- Shade the intervals \( (-\infty, \frac{2}{3}) \) and \( [\frac{3}{4}, \infty) \).

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.

Interval Notation
Interval notation is a mathematical shorthand to describe the set of solutions to an inequality. It efficiently captures all the values that a variable can take without listing each one individually. In our example with the inequality \( \frac{1}{3x-2} \leq 4 \), we found that the solution set is \( (-\infty, \frac{2}{3}) \cup [\frac{3}{4}, \infty) \). These intervals tell us that:
  • From negative infinity up to (but not including) \( \frac{2}{3} \), the inequality holds true.
  • There is a pause between \( \frac{2}{3} \) and \( \frac{3}{4} \), where the inequality does not hold.
  • Then, starting from \( \frac{3}{4} \) (inclusive), extending to positive infinity, the inequality is again true.
Here, parentheses \(( )\) indicate that the endpoint is not included, while square brackets \([ ]\) show that the endpoint is part of the solution. This efficient notation helps avoid long lists of values and makes it easier to understand the set of possible solutions quickly.
Critical Points
Critical points play a crucial role in solving inequalities. They are points on the number line where the inequality changes its sign. For a rational inequality like \( \frac{9 - 12x}{3x-2} \leq 0 \), critical points arise from setting both the numerator and the denominator equal to zero.
  • When the numerator of the fraction is zero, the entire fraction equals zero. In our case, \( 9 - 12x = 0 \) leads to the critical point \( x = \frac{3}{4} \).
  • When the denominator is zero, the expression is undefined. Here, \( 3x - 2 = 0 \) gives us \( x = \frac{2}{3} \).
The critical points help break down the number line into intervals. These are essential for testing where the inequality holds true, helping understand where the rational expression changes behavior.
Number Line Graph
Graphing solutions on a number line provides a visual representation of which segments satisfy an inequality. In equations like \( \frac{1}{3x-2} \leq 4 \), the number line graph distinctly shows where the inequality holds:
  • Place an open circle at \( \frac{2}{3} \) to indicate that it is not part of the solution. The inequality is undefined at this point.
  • Put a closed circle at \( \frac{3}{4} \) because the inequality equals zero here, meaning it's included in the solution.
To complete the graph, shade the regions \( (-\infty, \frac{2}{3}) \) and \( [\frac{3}{4}, \infty) \). These shaded areas show the continuous ranges of \( x \) values that make the initial inequality true. The graph serves as a quick reference guiding you through interval implications.
Rational Inequalities
A rational inequality involves quotients of polynomials, like \( \frac{1}{3x-2} \leq 4 \). These inequalities can be tricky due to the presence of variables in the denominators causing fractions to become undefined. To solve such inequalities, follow these steps:
  • Reorganize the inequality so one side is zero, as done with \( \frac{1}{3x-2} - 4 \).
  • Find critical points by making the numerator zero or the denominator zero, leading to potential changes in the inequality's sign.
  • Use test points in different intervals established by the critical points to check where the inequality holds.
Understanding rational inequalities helps manage expressions that behave differently based on the input values. It is important to handle them cautiously due to potential undefined points or jumps in behavior at critical points.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed in Problems 16-23. $$ y=3 \cos \frac{x}{2} $$

Convert the following radian measures to degrees. (a) \(\frac{7}{6} \pi\) (b) \(\frac{3}{4} \pi\) (c) \(-\frac{1}{3} \pi\) (d) \(\frac{4}{3} \pi\) (e) \(-\frac{35}{18} \pi\) (f) \(\frac{3}{18} \pi\)

A belt fits tightly around the two circles, with equations \((x-1)^{2}+(y+2)^{2}=16\) and \((x+9)^{2}+(y-10)^{2}=16\) How long is this belt?

In Problems \(29-34\), find an equation for each line. Then write your answer in the form \(A x+B y+C=0\). Through \((3,4)\) with slope \(-1\)

In Problems 45 -48, find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. \(5 x-2 y=5\) \(2 x+3 y=9\) \(2 x+3 y=6\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.