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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.

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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.

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Most popular questions from this chapter

Write an equation using the information given in the problem. Use \(x\) as the variable. Then solve the equation. If six times a number is subtracted from seven times the number, the result is \(-9 .\) Find the number.

Write the answer to each problem in terms of the variable. A football player gained \(x\) yards rushing. On the next down, he gained 9 yd. What expression represents the number of yards he gained altogether?

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When the lesser of two consecutive integers is added to three times the greater, the result is 43. Find the integers.

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