Chapter 8: Problem 64

Solve each inequality. State the solution set using interval notation when possible. \(9-x^{2}<0\)

### Short Answer

## Step by step solution

## Understand the inequality

## Rewrite the inequality

## Solve the inequality

## Write the solution in interval notation

## Key Concepts

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

###### quadratic inequalities

By rewriting the inequality \(9 - x^2 < 0\) as \(9 < x^2\), it becomes more apparent which regions of the x-axis need to be considered. This transformation allows us to identify the critical points, in this case, \(x = 3\) and \(x = -3\). It's these points where the parabola changes direction, touching the x-axis.

To solve \(9 < x^2\), you must determine the values for \(x\) that make \(x^2\) greater than 9. In this scenario, these values are where \(x > 3\) or \(x < -3\). Thus, the solution to the inequality includes these two regions on the x-axis.

###### interval notation

- Parentheses \(()\) indicate that the end value is not included in the interval (open interval).
- Brackets \([]\) signify that the end value is included in the interval (closed interval).

Therefore, to express this in interval notation, we write:

\( (-\rightarrow -3) \cup (3 \rightarrow \rightarrow) \)

In this notation, the symbol \(\cup\) represents the union of sets. It indicates that any value that lies in either of the two intervals (but not the values -3 and 3 themselves) satisfies the inequality.

###### polynomial expression

- Polynomial expressions are made up of terms, each term consisting of a coefficient (a number) multiplied by a variable raised to a power.
- The highest power of the variable is called the degree of the polynomial. In this case, the degree is 2, making it a quadratic polynomial.
- Quadratic polynomials usually produce parabolic graphs when plotted.

The terms and structure of the polynomial inform us how to approach and break down the problem into solvable steps.