/*! 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 50 Solve each inequality and graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 50

Solve each inequality and graph the solution set on a number line. \(4 x-7>9 x-2\)

Short Answer

Expert verified
The solution to the inequality is \(x < -1\). When graphed on a number line, an open circle is placed at -1 and an arrow points to the left of -1.

Step by step solution

01

Simplify the inequality

In order to simplify the inequality, group all terms involving \(x\) on one side and constant terms on the other. To do this, subtract \(4x\) from both sides and add \(2\) to both sides. This will result in: \(0 > 5x + 5\).
02

Continue simplifying

To get \(x\) alone on one side, subtract \(5\) from both sides of the inequality to get: \(-5 > 5x\).
03

Solve for x

To solve for \(x\), divide both sides of the inequality by \(5\). Note that we don't need to reverse the inequality sign because we're dividing by a positive number. This yields the solution: \(-1 > x\) or equivalently, \(x < -1\).
04

Graph the solution on a number line

Place a circle at \(-1\) on the number line to represent the boundary of the solution set. Since \(-1\) is not a solution to the inequality \(x <-1\), the circle should be open (not filled in). With \(x < -1\), the arrow should point to the left, representing all numbers less than \(-1\).

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.

Number Line
Using a number line can make understanding inequalities much easier. A number line is a simple graphic that visually displays numbers in order, helping us see how numbers relate to each other in terms of value. When graphing an inequality on a number line, there are a few important steps to follow.
  • Mark the point on the number line corresponding to the number mentioned in the inequality.
  • If the inequality is strict (like < or >), use an open circle to indicate that the number itself is not included in the solution. For example, in the inequality \( x < -1 \), a circle is placed around -1 but left open.
  • Draw an arrow to the left or right depending on whether the inequality is less than or greater than. The direction indicates the range of solutions.
This visual can help ensure you understand which numbers make the inequality true. By seeing it on a number line, learners can interpret and confirm the solution easily.
Solution Set
The solution set of an inequality refers to all the possible values that satisfy the condition of the inequality. After solving an inequality, it鈥檚 essential to express the solution clearly with a solution set.

For example, in the inequality \(4x - 7 > 9x - 2\), after simplifying, we find the solution \(x < -1\). This means any number less than -1 is part of the solution set. To clearly express this set:
  • Use interval notation, described as \((-\infty, -1)\), where \(-\infty\) indicates there is no end to the lower values.
  • Alternatively, set builder notation can be useful: \( \{x | x < -1\} \), showing the set of all \(x\) values where \(x\) is less than \(-1\).
Writing the solution set helps communicate precisely which values will satisfy the original inequality. Understanding how to read and write solution sets is crucial for ensuring that your solutions are complete and correct.
Simplifying Inequalities
Simplifying inequalities is crucial for finding the solution set efficiently. The process often involves rearranging terms and isolating the variable to discover its possible values.

To simplify the inequality \(4x - 7 > 9x - 2\), it is important to follow these steps:
  • Move all terms involving the variable \(x\) to one side. For example, subtract \(4x\) from both sides to get \(0 > 5x + 5\).
  • Move constant terms to the other side by subtracting \(5\) from each side, giving \(-5 > 5x\).
  • Solve for \(x\) by isolating it. This means dividing each side by 5, resulting in \(-1 > x\) or equivalently, \(x < -1\).
  • Be mindful when multiplying or dividing by negative numbers, as this requires flipping the inequality sign, although it is not needed in this example.
By simplifying inequalities effectively, you can make solving them straightforward and ensure accuracy in your solution.

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

Factor the trinomials or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(6 x^{2}-17 x+12\)

In Exercises 33-36, solve each equation using the zero-product principle. \((x-8)(x+3)=0\)

Solve the quadratic equations in Exercises 37-52 by factoring. \(5 x^{2}+x=18\)

Solve each equation by the method of your choice. \(7 x(x-2)=3-2(x+4)\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}-x=14\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Pure 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.