Problem 46 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ x<2 \text { and } x \leq-1 $$

Step-by-Step Solution

Answer

The intersection is all \( x \leq -1 \), represented as \((- fty, -1]\).

Step 1: Understanding the Inequalities

We have two inequalities to consider: \( x < 2 \) and \( x \leq -1 \). The solution to the problem will be where these two inequalities overlap on the number line.

Step 2: Solving Each Inequality

- For \( x < 2 \): The solution includes all numbers that are less than 2. On the number line, this is represented by an open dot at 2 extending leftward.- For \( x \leq -1 \): The solution includes all numbers that are less than or equal to -1. On the number line, this is represented by a closed dot at -1 extending leftward.

Step 3: Finding the Intersection

We now look for numbers that satisfy both inequalities simultaneously. Thus, we need the overlap (intersection) of the regions for both \( x < 2 \) and \( x \leq -1 \). The overlapping region is all numbers less than or equal to -1, represented with a closed dot at -1 extending to the left indefinitely.

Step 4: Graphing on the Number Line

Draw a number line and mark the point -1 with a closed dot, extending to the left with a line. Since the intersection covers all values \( x \leq -1 \), there is no need to extend to or past 2.

Step 5: Writing in Interval Notation

The interval notation for the values is \((-fty, -1]\), indicating all numbers less than or equal to -1, with -1 included in the interval.

Key Concepts

Number Line

Understanding how to use a number line can help visualize solutions to inequalities. A number line is a straight line with numbers placed at equal intervals along it. It allows us to see where numbers fall in relation to each other.

When dealing with inequalities like \( x < 2 \) and \( x \leq -1 \), we need to plot them on the number line.

• For \( x < 2 \), we place an open dot at 2 because 2 is not included in the solution; the line extends to the left.
• For \( x \leq -1 \), a closed dot is used at -1 because -1 is included in the solution; again, the line extends to the left.

Visualizing how these sections overlap helps find the solution to a system of inequalities.

Interval Notation

Interval notation is a simpler way to express a range of values that satisfy an inequality. It uses brackets to signify whether endpoints are included or not.

For the inequalities \( x < 2 \) and \( x \leq -1 \), you determine the intervals:

• \( x < 2 \) is expressed as \((-\infty, 2)\) because it goes up to but does not include 2.
• \( x \leq -1 \) is \((-\infty, -1]\) because it goes up to and includes -1.

The intersection of these intervals is \((-\infty, -1]\), representing all solutions common to both inequalities. Using interval notation makes it clear which numbers are part of the solution set.

Intersection of Inequalities

Finding the intersection of inequalities involves identifying values that satisfy all conditions simultaneously.

For the inequalities \( x < 2 \) and \( x \leq -1 \):

• Visually overlap the solutions on a number line to find the shared region.
• The intersection represents values that are solutions to both inequalities.
• For our problem, this is all numbers less than or equal to -1, visible as a closed segment at -1 extending indefinitely to the left.

This process reliably narrows solutions into a single, concise interval that can be universally applied to the problem at hand.