Problem 35 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ x \leq 1 \quad \text { or } \quad x \leq 8 $$

Step-by-Step Solution

Answer

The solution set is \( (-\infty, 8] \).

Step 1: Understand the given inequalities

The problem gives two inequalities: \( x \leq 1 \) and \( x \leq 8 \). The solution involves solving each inequality separately and then combining the results.

Step 2: Solve the first inequality

The first inequality is \( x \leq 1 \). This means that \( x \) can be any value that is less than or equal to 1.

Step 3: Solve the second inequality

The second inequality is \( x \leq 8 \). This means that \( x \) can be any value that is less than or equal to 8.

Step 4: Combine the solutions of both inequalities

Since the inequalities are combined with 'or', we need values that satisfy either or both conditions. This means the solution set includes all values \( x \leq 8 \) since any value that is less than or equal to 8 also includes all values less than or equal to 1.

Step 5: Write the solution in interval notation

In interval notation, the solution set is \( (-\infty, 8] \). This represents all values from negative infinity up to and including 8.

Step 6: Graph the solution set

On a number line, draw a solid circle at 8 and shade the line to the left extending to negative infinity. This represents all values less than or equal to 8.

Key Concepts

Solving Inequalities

To solve inequalities like the example given, you need to handle each part of the compound inequality separately. In our example, we have:
\( x \leq 1 \) and \( x \leq 8 \).
The goal is to identify all values of x that satisfy each inequality. For the first inequality, \( x \leq 1 \), this means x can be any value from negative infinity up to and including 1.
For the second inequality, \( x \leq 8 \), this means x can be any value from negative infinity up to and including 8.
Since these inequalities are combined with 'or', our final solution will be the union of the two individual solutions.
This essentially means that any value meeting either criterion will be included in our final solution.

Interval Notation

Interval notation is a way to represent the set of solutions to an inequality. In our problem, we are combining the solutions \( x \leq 1 \) and \( x \leq 8 \) using 'or'.
Since the solution for \( x \leq 8 \) naturally includes the solution for \( x \leq 1 \), the overall solution is:

• x is in the interval \(( - \infty, 8 ] \)

This means x can take any value from negative infinity up to and including 8. The parentheses indicate that negative infinity is not a specific number but a concept, and the bracket means 8 is included in the solution set.
This concise way of writing the solution helps to clearly communicate the entire range of possible values satisfying the compound inequality.

Graphing Solutions

Graphing the solution to an inequality on a number line provides a visual representation of all the values x can take. For the inequality \( x \leq 8 \), here's how to graph it:

• Draw a number line.
• Find and mark the number 8 on the line with a solid circle. A solid circle indicates that this number is included in the solution.
• Shade the line to the left of 8, extending towards negative infinity. This shading represents all numbers less than or equal to 8.

The solid circle at 8 and the shading to the left effectively show that all values from negative infinity up to and including 8 are part of the solution.
This graphical method complements interval notation and provides another way to understand the solution range.