Problem 19 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ x<2 \text { and } x>-3 $$

Step-by-Step Solution

Answer

(-3, 2)

Step 1 - Understand Each Inequality

Analyze the given inequalities separately. The first inequality is given by: x < 2. This means that x is any number less than 2.

Step 2 - Analyze the Second Inequality

The second inequality is: x > -3. This means that x is any number greater than -3.

Step 3 - Combine the Inequalities

Since it is given that x must satisfy both inequalities at the same time (x < 2 and x > -3), the solution set is the intersection of these two inequalities. Thus, -3 < x < 2.

Step 4 - Write the Solution in Interval Notation

The combined solution set -3 < x < 2 in interval notation is written as: (-3, 2).

Step 5 - Graph the Solution Set

Graph the interval (-3, 2) on a number line. Draw an open circle at -3 and an open circle at 2, and shade the region between them to indicate all numbers x that satisfy -3 < x < 2.

Key Concepts

solving inequalities

Solving inequalities involves finding the set of all possible values that satisfy the given inequality. An inequality is a mathematical expression that shows the relationship between two values that are not necessarily equal. For example, in the inequality \(x < 2\), the values of \(x\) are all less than 2.

Here's a quick guide on how to solve inequalities:

• First, understand each inequality separately.
• Next, combine the inequalities if they form a compound statement.
• Finally, express the combined solution if necessary.

When working on compound inequalities like \(x < 2\) and \(x > -3\), you need to find the common values that satisfy both conditions. In this case, combining the two results in the compound inequality \(-3 < x < 2\).

The next steps involve expressing and graphing the solution set correctly.

interval notation

Interval notation is a compact way of describing a range of numbers that make up the solution set of an inequality. It's very useful for representing solutions to inequalities.

Here’s how interval notation works:

• The parentheses \(()\) indicate that an endpoint is not included in the interval (known as an 'open' endpoint).
• The brackets \([]\) indicate that an endpoint is included (known as a 'closed' endpoint).

For the compound inequality \(-3 < x < 2\), the solution set includes all values between \(-3\) and \(2\), but not \(-3\) or \(2\) themselves. This is written in interval notation as \((-3, 2)\).

Using interval notation makes it easier to read and understand the solution set at a glance.

graphing inequalities

Graphing inequalities provides a visual representation of the solution set on a number line. This helps in understanding the range of values that satisfy the inequality.

To graph the solution set of \(-3 < x < 2\):

• First, draw a number line.
• Mark the points \(-3\) and \(2\) on the number line.
• Use open circles at \(-3\) and \(2\) to indicate that these values are not included in the solution set.
• Shade the area between \(-3\) and \(2\) to show that all numbers in this range satisfy the inequality.

This visual representation makes it clear that any number between -3 and 2 is a solution to the given compound inequality.

Graphing not only clarifies the solution but also assists in understanding how different types of inequalities work together to define a range of solutions.