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Chapter 2: Problem 80

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ -3 \geq-\frac{1}{3} t $$

Short Answer

Expert verified
The solution is \([9, \infty)\).

Step by step solution

01

Isolate the variable

We need to isolate the variable \( t \) on the right side of the inequality \(-3 \geq -\frac{1}{3} t\). Start by dividing both sides of the inequality by \(-\frac{1}{3}\). Remember that dividing by a negative number reverses the inequality sign.
02

Divide and reverse the inequality sign

Divide both sides of the inequality by \(-\frac{1}{3}\):\[ -3 \div \left(-\frac{1}{3}\right) \leq t \] This simplifies to: \[ 9 \leq t \] We can rewrite it as: \[ t \geq 9 \]
03

Write the solution in interval notation

The inequality \( t \geq 9 \) means that \( t \) is greater than or equal to 9. In interval notation, this is written as \([9, \infty)\).
04

Graph the solution

To graph the solution \([9, \infty)\), draw a number line. Place a closed circle at 9 and shade the line to the right of 9 to indicate all numbers greater than or equal to 9.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Inequalities
Solving inequalities is a lot like solving equations, with a few extra rules to keep in mind. The goal is to isolate the variable, meaning you want the variable (like \( t \) in our example) on one side of the inequality sign by itself. Let's break down how to solve inequalities easily:
  • Perform the same operations on both sides: Just like with equations, you can add, subtract, multiply, or divide both sides of an inequality by the same number.
  • Watch out for negatives: If you multiply or divide both sides of an inequality by a negative number, remember to flip the direction of the inequality sign. This is crucial!
For example, with the inequality \( -3 \geq -\frac{1}{3} t \), we divided both sides by \(-\frac{1}{3}\). Because the division involved a negative number, we flipped \( \geq \) to \( \leq \), which gave us the solution \( t \geq 9 \). This tells us that any value for \( t \) must be 9 or more.
Interval Notation
Interval notation is a way to represent sets of numbers, particularly useful when dealing with solutions to inequalities. It shows which numbers are included in a solution set and can provide a quick visual reference:
  • Use brackets \([ \) or \( ] \) to show inclusive endpoints: If a number is part of the solution, use square brackets to include it.
  • Use parentheses \(( \) or \() \) for exclusive endpoints: If a number isn't included, such as infinity, use parentheses.
For our solution \( t \geq 9 \), we write the interval notation as \([9, \infty)\). This indicates that 9 is part of the solution, and the set continues to infinity. Infinity always gets a parenthesis because it's a concept, not a number, and hence, can never be included.
Graphing Inequalities
Graphing inequalities provides a visual representation of the solution set on a number line, which is very helpful for understanding the range of possible solutions. Here's how to graph inequalities like \( t \geq 9 \):
  • Draw a number line: Start with a straight, horizontal line and mark the relevant points.
  • Use circles to denote endpoints: Place a closed circle on the number line at the point 9 to show it's included in our set.
  • Shade the solution area: Since \( t \) can be greater than or equal to 9, shade the line to the right of 9. This shading shows all the values \( t \) can take.
By visualizing the inequality, you reinforce the solution's understanding. It's easy to see that any number to the right of 9 satisfies the inequality \( t \geq 9 \). This method offers a clear and quick way to grasp which numbers are solutions.

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Most popular questions from this chapter

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