Problem 21 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ x \leq 2 \text { and } x \leq 5 $$

Step-by-Step Solution

Answer

The solution set is \( (-\infty, 2] \).

Step 1: Understand the Compound Inequality

The compound inequality is given as: 1. \( x \leq 2 \) 2. \( x \leq 5 \) Both inequalities must be true at the same time.

Step 2: Find the Intersection

Since both conditions \( x \leq 2 \) and \( x \leq 5 \) must be true, we need to find the smallest value where both conditions overlap. Any value for \( x \leq 2 \) will also satisfy \( x \leq 5 \). Therefore, the solution is determined by the stricter condition, \( x \leq 2 \).

Step 3: Write the Solution in Interval Notation

The interval notation for \( x \leq 2 \) is \( (-\infty, 2] \).

Step 4: Graph the Solution on a Number Line

Draw a number line, and shade all the points to the left of and including 2: ----|----•----- \(-\infty, 2] \)Highlight the point at 2 with a solid filled circle to show it is included in the solution.

Key Concepts

Compound Inequalities

Compound inequalities are expressions that combine two or more inequalities involving the same variable. You can see statements connected by words like 'and' or 'or'. The solution to a compound inequality depends on whether you are dealing with 'and' (intersection) or 'or' (union) inequalities.

• 'And' inequalities: Both conditions must be true simultaneously. You consider the overlap of the solution sets.
• 'Or' inequalities: At least one of the conditions must be true. You combine the solution sets.

In the given exercise, we have:
$$ x \ leq 2 \text { and } x \ leq 5 $$
This is an 'and' inequality, meaning we need to find where both conditions overlap. Since any value satisfying
$$ x \ leq 2 $$
also satisfies
$$ x \ leq 5 $$
our solution is determined by the stricter condition:
$$ x \ leq 2.$$

Inequality Interval Notation

Interval notation is a way to represent the solution set of an inequality. It uses intervals to describe where the variable values lie on the number line. Here's how it works:

• Open interval: Parentheses, ( ), indicate that the endpoint is not included in the solution.
• Closed interval: Brackets, [ ], indicate that the endpoint is included.

For instance, in our example,
$$ x \ leq 2 $$
the solution includes all real numbers less than or equal to 2. This is noted as
\begin{align*} (-\infty, 2] \ \ end{align*}

Graphing Inequalities

Graphing inequalities on a number line helps visualize the solution set. Here's how to graph them step-by-step:
1. Draw a number line.
2. Identify and mark the endpoint. For
$$ x \ leq 2 $$
, place a solid circle at 2 since the inequality is '<='.
3. Shade the region that represents the solution set. In this case, shade everything to the left of 2. The solid circle at 2 indicates that 2 is part of the solution.
The final graph looks like this:
\begin{align*} ----|----•---- \ end{align*}
To summarize, graphing provides a clear way to see and confirm the solution. When combined with interval notation, it gives a thorough understanding of the solution set of an inequality.