/*! 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 48 Solve \(5(x+1) \leq 2(x-3) .\) W... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 48

Solve \(5(x+1) \leq 2(x-3) .\) Write the solution set in interval notation and graph it.

Short Answer

Expert verified
The solution set is \((-\infty, \frac{-11}{3}]\).

Step by step solution

01

Distribute Terms

Distribute the constants outside the parentheses on both sides of the inequality: For the left side: \[ 5(x+1) = 5x + 5 \]For the right side: \[ 2(x-3) = 2x - 6 \]
02

Formulate the Inequality

Substitute the distributed expressions back into the inequality: \[ 5x + 5 \leq 2x - 6 \]
03

Isolate the Variable

Subtract \(2x\) from both sides to bring the terms involving \(x\) to one side: \[ 5x - 2x + 5 \leq -6 \]This simplifies to:\[ 3x + 5 \leq -6 \]Now, subtract 5 from both sides:\[ 3x \leq -11 \]
04

Solve for x

Divide each side of the inequality by 3 to solve for \(x\):\[ x \leq \frac{-11}{3} \]
05

Write the Solution in Interval Notation

The solution \( x \leq \frac{-11}{3} \) can be written in interval notation as:\[ (-\infty, \frac{-11}{3}] \]
06

Graph the Solution Set

To graph the solution set, draw a number line and shade the region to the left of \(\frac{-11}{3}\). Place a closed circle at \(\frac{-11}{3}\) to indicate that \(\frac{-11}{3}\) is included in 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.

Interval Notation
Interval notation provides a concise way to express a range of values. It is especially useful in inequalities where the solution is not just a single number but a set of numbers. For example, when we solve the inequality \( x \leq \frac{-11}{3} \), it indicates that \( x \) can be any number less than or equal to \( \frac{-11}{3} \). Here is how interval notation comes into play:
  • If a number is included in the solution set, it is represented with a square bracket \([ ) or ( ]\).
  • If a number is *not* included, it is represented with a parenthesis \(( ) or )\).
  • An infinity symbol \( \infty \) represents unbounded intervals.
So for our solution, \( (-\infty, \frac{-11}{3}] \) signifies all numbers from negative infinity up to and including \( \frac{-11}{3} \). The parenthesis around \( -\infty \) shows that infinity is not a reachable limit, while the bracket around \( \frac{-11}{3} \) includes this endpoint.
Graphing Solutions
Graphing solutions to inequalities helps in visualizing the range of possible values for a variable. For the inequality \( x \leq \frac{-11}{3} \), start by drawing a number line.Here鈥檚 how to graph the solution step-by-step:
  • Identify the point \( \frac{-11}{3} \) on the number line. This is your critical value.
  • Place a closed circle at \( \frac{-11}{3} \) because the inequality includes this number. A closed circle means "including" the point.
  • Shade the number line to the left of \( \frac{-11}{3} \) to indicate that all numbers less than \( \frac{-11}{3} \) are part of the solution set.
Using this method, you can clearly see the range of solutions and understand the concept of inequalities more intuitively.
Distributive Property
The distributive property is a key concept for solving equations and inequalities efficiently. It allows you to multiply a single term by each term inside a parenthesis. In our exercise, we used the distributive property to expand the expressions \( 5(x+1) \) and \( 2(x-3) \).Here鈥檚 how the distributive property works:
  • Multiply the term outside the parenthesis by each term within the parenthesis.
  • For \( 5(x+1) \): Distribute the \( 5 \) as \( 5 \times x + 5 \times 1 = 5x + 5 \).
  • For \( 2(x-3) \): Distribute the \( 2 \) as \( 2 \times x + 2 \times (-3) = 2x - 6 \).
Understanding this property is crucial since it simplifies expressions, making it easier to isolate variables and solve equations or inequalities. It鈥檚 a foundational concept in algebra.

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

Why is \(y-y_{1}=m\left(x-x_{1}\right)\) called the point-slope form of the equation of a line?

Find the slope of each line. See Examples 4 and \(5 .\) $$x+14=0$$

$$ \text { If } f(x)=x^{2}+x, \text { find: } f\left(\frac{4}{5} r\right) $$

College Fees. \(\quad\) Each semester, students enrolling at a community college must pay tuition costs of \(\$ 20\) per unit as well as a \(\$ 40\) student services fee. a. Write a linear equation that gives the total fees \(t\) to be paid by a student enrolling at the college and taking \(x\) units. b. Use your answer to part a to find the enrollment cost for a student taking 12 units.

Streets. One of the steepest streets in the United States is Eldred Street in Highland Park, California (near Los Angeles). It rises approximately 220 feet over a horizontal distance of 665 feet. What is the grade of the street? (IMAGE CANNOT COPY)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.