/*! 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 55 Solve the inequality. Express yo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 55

Solve the inequality. Express your answer in interval notation. $$-4(x+2) \geq x+5$$

Short Answer

Expert verified
\((-鈭�, -2.6]\)

Step by step solution

01

Distribute

First, distribute the -4 inside the parentheses to obtain: \(-4x - 8 \geq x + 5\).
02

Collect Like Terms

Next, move terms involving x to one side of the inequality and numbers to the other side in order to collect like terms. Subtract x from each side to get: \(-5x - 8 \geq 5\). Then, add 8 to each side: \(-5x \geq 13\).
03

Solve for x

Divide both sides of the inequality by -5 to solve for x, remembering that when dividing by a negative number, the direction of the inequality changes: \(x \leq -13/5\) or \(x \leq -2.6\).
04

Write in Interval Notation

The solution in interval notation would be written as: \((-鈭�, -2.6]\).

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 method used to specify the set of solutions for an inequality. It's a concise way to express which numbers are included in a set. In interval notation:
  • A round bracket, "(", is used when a number is not included, meaning it is an open interval.
  • A square bracket, "]", is used when a number is included in the set, marking a closed interval.
  • The symbols "鈭�" and "-鈭�" are used to represent unbounded ends of an interval, signifying no limits in that direction.

For example, in the solution \((-鈭�, -2.6]\), this tells us that all numbers less than or equal to -2.6 are part of the solution set. The "(" symbol before -鈭� shows that there is no lower bound on the interval, and the "]" around -2.6 indicates -2.6 is included.
Distributive Property
The distributive property is a fundamental principle in algebra that allows you to simplify expressions by expanding terms. When applying the distributive property, you multiply a term outside the brackets by each term inside the brackets. Mathematically, it is represented as:\[ a(b + c) = ab + ac \]In the given inequality \-4(x+2) 鈮� x+5\, you apply this property by multiplying \-4\ by each term inside the parentheses. This results in \-4x - 8\.
Using this property can help in breaking down expressions into simpler parts, making them easier to solve or manipulate in equations and inequalities. It鈥檚 widely used in both simple and complex algebraic problems.
Solving Linear Inequalities
Solving linear inequalities is a process akin to solving equations, but it involves finding a range of values satisfying the inequality rather than a single value. Here are the general steps when solving inequalities:
  • Distribute and simplify: Use distributive property if necessary and simplify the expression.
  • Move terms: Gather all variables on one side of the inequality and constants on the other.
  • Isolate the variable: Use arithmetic operations to isolate the variable. Remember, if you multiply or divide both sides of an inequality by a negative number, the inequality symbol must be flipped.
  • Express the solution: Once the variable is isolated, express the solution in interval notation.

Taking these steps, as shown in the exercise, leads to the solution \(x \leq -2.6\), expressed in interval notation as \((-鈭�, -2.6]\). This shows understanding not just of finding solutions, but how they are represented mathematically.

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

Solve the inequality algebraically and graphically. Express your anstoer in intereal notation. $$2 x<3 x-10$$

A telephone company offers two different long-distance calling plans. Plan A charges a fee of S4. 95 per month plus \(\$ 0.07\) for each minute used. Plan B costs \(\$ 0.10\) per minute of use, but has no monthly fee. (IMAGE CANNOT COPY) (a) Find the total monthly cost of using Plan \(A\) as a linear function of the number of minutes used. (b) Find the total monthly cost of using Plan \(B\) as a linear function of the number of minutes used. (c) Interpret the \(y\) -intercept of the graph of each cost function. (d) Calculate algebraically the number of minutes of long-distance calling for which the two plans will cost the same. What will be the monthly charge at that level of usage? 4 (e) \(\quad\) Graph the functions from parts (a) and (b) on the same set of axes and find the number of minutes of long-distance calling for which the two plans will cost the same. You will have to adjust the window size and scales appropriately. What is the monthly cost at that level of usage? Compare your result with the result you found algebraically.

Pricing Tickets Sherman is planning to bring in a jazz group of four musicians for a fund-raising concert at Grand State University. The jazz group charges \(\$ 500\) for an appearance, and dinner will be provided to the musicians at a cost of \(\$ 20\) each. In addition, the musicians will be reimbursed for mileage at a rate of \(\$ 0.30\) permile. The group will be traveling a total of 160 miles. A ticket for the concert will be priced at \(\$ 8 .\) How many people must attend the concert for the university to break even?

Solve the inequality. Express your answer in interval notation. $$0<-x+5<4$$

Travel This problem is an extension of Example \(1 .\) A one-way ticket on a weckday from Newark, New Jersey, to New York, New York, costs 3.30 dollars for a train departing during peak hours and 2.50 dollars for a train departing during off-peak hours. Peak morning hours are from 6 A.M. to 10 A.M. and peak evening hours are from 4 P.M. to 7 P.M. The rest of the day is considered to be off-peak. (Source: New Jersey Transit) (a) Construct a table that takes the time of day as its input and gives the fare as its output. (b) Write the fare as a function of the time of day using piecewise function notation. (c) Graph the function.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.