Problem 28 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ x \leq-20 \quad \text { or } x \geq-10 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -20] \cup [-10, \infty)\).

Step 1: Identify the Inequality

The inequalities given in the exercise are: \( x \leq -20 \) and \( x \geq -10 \). We will solve and graph them individually on the number line to understand the solution sets.

Step 2: Graph the First Inequality

For \( x \leq -20 \), shade the number line to the left of \(-20\) and use a closed circle at \(-20\) since the inequality includes \(-20\).

Step 3: Graph the Second Inequality

For \( x \geq -10 \), shade the number line to the right of \(-10\) and use a closed circle at \(-10\) since the inequality includes \(-10\).

Step 4: Combine the Graphs

Since this is an 'or' inequality, combine both graphs. This means we include all the shaded regions from both inequalities as part of the solution.

Step 5: Write Interval Notation

For \( x \leq -20 \), the interval notation is \((-\infty, -20]\). For \( x \geq -10 \), the interval notation is \([-10, \infty)\). Combining them, the solution in interval notation is \((-\infty, -20] \cup [-10, \infty)\).

Key Concepts

Understanding the Number Line

The number line is a visual tool used in math to represent numbers in order. It helps to understand relationships quickly.
When dealing with inequalities, the number line becomes crucial. Placing numbers on it shows how one number relates to another. Negative numbers are to the left, positive to the right.
For inequalities like \( x \leq -20 \) and \( x \geq -10 \), we use circles:

• Closed Circle: This indicates that a number is included. For \( x \leq -20 \), we place a closed circle at -20.
• Shaded Region: This shows where we shade the line to represent all numbers included. With \( x \leq -20 \), we shade left from -20.

By seeing the graph on a number line, it’s simple to visualize solutions.

Exploring Interval Notation

Interval notation provides a quick and compact way to express the range of numbers in a solution set.
It uses brackets and parentheses to describe intervals:

• Parentheses \( ( ) \): These show that a number is not included. We use them with infinity, like \((-\infty\).
• Brackets \( [ ] \): These show inclusion. For \( x \leq -20 \), write \([-20]\), meaning -20 is included.

For the example \( x \leq -20 \) or \( x \geq -10 \), we find:

• \((-\infty, -20]\): All numbers less than or equal to -20.
• \([-10, \infty)\): All numbers greater than or equal to -10.

The two intervals can be combined with a union \( \cup \): \((-\infty, -20] \cup [-10, \infty)\).

Graphing Inequalities

Graphing inequalities involves drawing solutions onto a number line. This gives a clear picture of the range of possible values.
For \( x \leq -20 \):

• Place a closed circle at -20 to include it.
• Shade everything to the left, indicating numbers less than or equal to -20.

For \( x \geq -10 \):

• Place a closed circle at -10 to include it.
• Shade everything to the right, indicating numbers greater than or equal to -10.

These kinds of graphs help you understand where solutions overlap or diverge. It’s a simple step-by-step process that makes abstract inequalities more concrete and easy to grasp.