/*! 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 71 Solve inequality and graph the s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 71

Solve inequality and graph the solution set. \(3 x-(4-2 x)<5-(2-5 x)\)

Short Answer

Expert verified
The solution is all real numbers.

Step by step solution

01

Distribute and Simplify

First, simplify the inequality by distributing and combining like terms. Start with the original inequality:\(3x - (4 - 2x) < 5 - (2 - 5x)\)Distribute the negative signs:\(3x - 4 + 2x < 5 - 2 + 5x\)Combine like terms on both sides:\(5x - 4 < 3 + 5x\)
02

Isolate the Variable

Next, try to isolate the variable (x) on one side of the inequality. Subtract \(5x\) from both sides:\(5x - 5x - 4 < 3 + 5x - 5x\)This simplifies to:\(-4 < 3\)
03

Interpret the Result

After isolating the variable, interpret the inequality. The simplified statement \(-4 < 3\) is always true. Hence, the solution is all real numbers.
04

Graph the Solution Set

Since the inequality is always true, the solution set includes all real numbers. Represent this on a number line by shading the entire line, indicating that any value of \(x\) satisfies the inequality.

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.

Distributive Property
The distributive property helps us eliminate parentheses in algebraic expressions. It states that for any numbers \(a\), \(b\), and \(c\), the expression \(a(b + c) = ab + ac\). In the given inequality, we deal with parentheses: \(3x - (4 - 2x) < 5 - (2 - 5x)\). To apply the distributive property, distribute the negative signs inside the parentheses as follows: \(3x - 4 + 2x < 5 - 2 + 5x\). Always ensure to multiply each term inside the parenthesis by the factor outside before proceeding.
Combining Like Terms
Combining like terms means merging terms that have the same variables raised to the same power. In our simplified expression \(3x - 4 + 2x < 5 - 2 + 5x\). We can merge \(3x\) and \(2x\) on the left side to get \(5x - 4 \), and simplify the constants on the right side to get \(3\), so the inequality becomes \(5x - 4 < 3 + 5x\). This step makes the equation easier to handle in subsequent steps.
Isolation of Variables
To solve an inequality, we need to isolate the variable on one side. For our current inequality \(5x - 4 < 3 + 5x\), subtract \(5x\) from both sides to isolate \(x\). This gives us \(5x - 5x - 4 < 3 + 5x - 5x\), simplifying to \(-4 < 3\). Since the variable \(x\) has been eliminated and the resulting statement \(-4 < 3\) is true, the solution involves all real numbers.
Graphing Inequalities
Graphing inequalities visually represent the solution set on a number line or coordinate plane. For our problem, the resulting inequality \(-4 < 3\) is always true, meaning all numbers satisfy the inequality. Thus, the solution set includes all real numbers. To graph this, shade the entire number line, indicating every possible value of \(x\). This visual helps in understanding the range of solutions better.

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

Show a complete solution to each problem. Alcohol solutions. Amy has two solutions available in the laboratory, one with \(5 \%\) alcohol and the other with \(10 \%\) alcohol. How much of each should she mix together to obtain 5 gallons of an \(8 \%\) solution?

Show a complete solution to each problem. Increasing the percentage. A mechanic has found that a car with a 16 -quart radiator has a \(40 \%\) antifreeze mixture in the radiator. She has on hand a \(70 \%\) antifreeze solution. How much of the \(40 \%\) solution would she have to replace with the \(70 \%\) solution to get the solution in the radiator up to \(50 \% ?\)

Solve each equation. Practice combining some steps. Look for more efficient ways to solve each equation. $$$5 x+1=0$$

Show a complete solution to each problem. Tennis. The distance from the bascline to the service line on a tennis court is 3 feet less than the distance from the service line to the net. If the distance from the bascline to the net is 39 feet, then what is the distance from the service line to the net?

Identify the variable and write an inequality that describes situation. Fred must have at least a 3.2 grade point average.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.