/*! 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 30 Solve inequality. Write the solu... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 30

Solve inequality. Write the solution set in interval notation, and graph it. \(6 m \geq-24\)

Short Answer

Expert verified
The solution set in interval notation is \( [-4, \infty) \).

Step by step solution

01

Understand the Inequality

Recognize that the inequality given is a simple linear inequality: \( 6m \geq -24\).
02

Isolate the Variable

Divide both sides of the inequality by 6 to solve for \( m \): \[ \frac{6m}{6} \geq \frac{-24}{6} \]Simplify to get: \[ m \geq -4 \]
03

Write the Solution in Interval Notation

The solution set for \( m \geq -4 \) includes all real numbers greater than or equal to -4. In interval notation, this is written as: \[ [-4, \infty) \]
04

Graph the Solution Set

On a number line, shade the region starting from -4 to positive infinity. Use a closed dot at -4 to indicate that -4 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.

Understanding Linear Inequalities
Linear inequalities involve expressions that use inequality symbols such as \(<, \, >, \, \leq, \, \geq \). They are similar to linear equations, but instead of an equals sign, they use inequality signs to show a range of possible solutions.

For example, the inequality 6m \geq -24 indicates that 6 times a variable m is greater than or equal to -24. The goal is to solve for m in order to determine which values make the inequality true.

To solve a linear inequality:
  • Isolate the variable, just like solving an equation.
  • Remember that if you multiply or divide by a negative number, you must reverse the inequality sign.
  • Represent the solution set either on a number line or using interval notation.

In the example given, we isolate m by dividing both sides by 6, resulting in m \geq -4. This means m can be any number greater than or equal to -4.
Interval Notation for Inequalities
Interval notation offers a compact way of representing solution sets for inequalities. It uses brackets and parentheses to indicate which numbers are included in the range.

Let's understand the symbols:
  • '[' or ']' : Including the endpoint (closed interval).
  • '(' or ')' : Excluding the endpoint (open interval).
  • \(\infty\) and -\(\infty\) : Represent infinity and negative infinity. These are always paired with parentheses since infinity is not a specific number that can be included.

For our example, m \geq -4 includes all numbers from -4 to positive infinity. In interval notation, this is shown as \([-4, \,\infty)\). This tells us -4 is included, but the set extends endlessly in the positive direction.
Graphing Inequalities on a Number Line
Graphing inequalities visually represents the solution set on a number line. Here's how you do it:

1. Draw a number line and mark relevant points.
2. Use a closed dot to indicate the number is included in the solution set (like \-4 in our example).
3. Shade the region where the solutions lie.
  • If the inequality is \(>\) or \(\geq\), shade to the right of the dot.
  • If it is \( < \) or \( \leq \), shade to the left.

In our case, for m \geq -4 , you mark -4 on the number line with a closed dot and shade from -4 to the right, extending to infinity. This shaded region shows all possible values of m that satisfy the inequality.

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 inequality. Write the solution set in interval notation, and graph it. \(-12 \leq \frac{1}{2} z+1 \leq 4\)

Write inequality in interval notation, and graph the interval. \(6 \geq x \geq-4\)

Solve each equation. $$ 0.06 x=300 $$

For a certain provider, an international phone call costs \(\$ 2.00\) for the first 3 min, plus \(\$ 0.30\) per minute for each minute or fractional part of a minute after the first 3 min. If \(x\) represents the number of minutes of the length of the call after the first 3 min, then \(2+0.30 x\) represents the cost of the call. If Alan Lebovitz has \(\$ 5.60\) to spend on a call, what is the maximum total time he can use the phone?

Translate statement into an inequality. Use \(x\) as the variable. Tracy could spend at most \(\$ 20\) on a gift.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Statistics

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.