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Chapter 14: Problem 20

Solve and write the answer in interval notation. $$3 n-9 \geq 2 n-8$$

Short Answer

Expert verified
The solution in interval notation is \([1, \infty)\).

Step by step solution

01

Simplifying the inequality

To simplify the inequality, subtract \(2n\) from both sides of the inequality to get \(n-9 \geq -8\).
02

Solving for n

Then, add 9 to both sides of the inequality to isolate \(n\) on the left side. This results in \(n \geq 1\).
03

Writing the answer in interval notation

Finally, the solution to the inequality \(n \geq 1\) in interval notation is \([1, \infty)\). This means \(n\) is greater than or equal to 1 and could take any values up to positive infinity.

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Key Concepts

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

Interval Notation
Interval notation is a way of representing a set of numbers along a number line. It's a concise form to show where solutions lie. In our example, the solution \(n \geq 1\) means \(n\) can be any number from 1 to infinity. We use brackets to indicate whether endpoints are included (closed interval) or not (open interval):
  • \([a, b]\) means \(a\) and \(b\) are included in the set.
  • \((a, b)\) means \(a\) and \(b\) are not included.
For \(n \geq 1\), we use the interval \([1, \infty)\):
  • A square bracket \([\) at 1 includes it in the set.
  • A parenthesis \(()\) at infinity \((\infty)\) means infinity is not a specific number, so it鈥檚 never included.
Simplifying Inequalities
Simplifying inequalities involves making them easier to solve by using basic algebraic operations. Our initial inequality is \(3n - 9 \geq 2n - 8\).
  • First, eliminate the \(2n\) from the right by subtracting \(2n\) from both sides, which simplifies the inequality to \(n - 9 \geq -8\).
  • Next, add 9 to both sides to further simplify and isolate the variable \(n\). This results in \(n \geq 1\).
The goal is to have the variable isolated on one side, making it straightforward to read the inequality and find solutions.
Solving Linear Inequalities
Solving linear inequalities is similar to solving linear equations, but with one key difference: the inequality sign. For \(3n - 9 \geq 2n - 8\), we aim to find all values of \(n\) that satisfy this condition.
  • Isolate the variable on one side, which we've done to get \(n \geq 1\).
  • Use operations that maintain the inequality, like adding or subtracting the same value from both sides.
  • Remember, if you multiply or divide by a negative number, flip the inequality sign.
The final solution, \(n \geq 1\), tells us the values satisfying the condition start from 1 and extend to infinity. Use interval notation to simplify this: \([1, \infty)\).

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Most popular questions from this chapter

Assume that \(n\) and \(a\) are both positive numbers. State whether the solution set of an inequality of the given form contains only negative numbers, only positive numbers, or both negative and positive numbers. $$x+na$$

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