Problem 35 - Calculus Single Variable

Question

In Exercises \(35-43,\) sketch the set on a real number line. \(\left\\{x: x>-2\right.\) and \(\left.x^{2}<9\right\\}\)

Step-by-Step Solution

Answer

The solution is the interval \((-2, 3)\).

Step 1: Identify the Inequalities

We have two inequalities: 1. \( x > -2 \) 2. \( x^2 < 9 \). We need to determine the intervals where these inequalities overlap.

Step 2: Solve the Inequality \( x^2 < 9 \)

Solve the inequality \( x^2 < 9 \) by taking the square root on both sides. This gives \( -3 < x < 3 \). Thus, the numbers \( x \) that satisfy this inequality are between \(-3\) and \(3\).

Step 3: Find the Intersection of the Inequalities

We now find the intersection of the intervals from the inequalities: - From \( x > -2 \), we have the interval \((-2, \infty)\).- From \( -3 < x < 3 \), we have the interval \((-3, 3)\).The intersection of these intervals is \((-2, 3)\).

Step 4: Sketch the Set on a Real Number Line

On a real number line, draw an open interval starting from \(-2\) to \(3\). Use open circles at \(-2\) and \(3\) to indicate that these points are not included in the set.

Key Concepts

Real Number Line

The real number line is a visual representation of the set of all real numbers.Imagine a straight horizontal line that stretches infinitely to the left and right.Each point on this line corresponds to a real number, including fractions, whole numbers, and irrational numbers like \( \sqrt{2} \).
Let's break it down further:

• The left side of the line represents negative numbers, getting more negative as you move further left.
• The right side represents positive numbers, getting larger as you move right.
• The center point, often marked as zero, divides the negative and positive sides.

To represent a set of numbers on the real number line, like the solution to an inequality, we use intervals.An interval is simply a part of the number line containing all the numbers between two endpoints.This is an important concept because it allows us to graphically show where solutions to inequalities lie.

Intersection of Intervals

When working with inequalities, we often need to consider where two or more conditions overlap.This overlap is called the intersection of intervals.
For instance, let's look at two intervals: \( (-3, 3) \) and \( (-2, \infty) \).

• \((-3, 3)\) includes all numbers between \(-3\) and \(3\), but does not include the endpoints \( -3 \) or \( 3 \).
• \((-2, \infty)\) includes all numbers greater than \(-2\), extending infinitely in the positive direction.

The intersection of these intervals, \((-2, 3)\), includes the numbers common to both of them.It's where both conditions are true.On a number line, this is the portion where the two intervals overlap.

Quadratic Inequalities

Quadratic inequalities involve expressions like \( x^2 < 9 \).These are similar to quadratic equations but use inequality symbols such as \(<\), \(>\), \(\leq\), or \(\geq\) instead of an equal sign.
To solve quadratic inequalities, follow these steps:

• First, treat the inequality as an equation and solve for the roots (e.g., \( x^2 = 9 \)). In this case, the roots are \( x = 3 \) and \( x = -3 \).
• These roots divide the number line into regions. Test a value from each region in the original inequality to see if it satisfies the inequality.
• The solution is the set of regions that satisfy the inequality. For \( x^2 < 9 \), the solution is \( -3 < x < 3 \).

Quadratic inequalities are a bit trickier than linear ones, but with practice, they'll become easier to handle.Graphing the solution on a number line can help visualize the valid regions.