Problem 47 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ x>0 \text { and } x \geq-1 $$

Step-by-Step Solution

Answer

Graph shows part right of 0; interval: \((0, \infty)\).

Step 1: Analyze the Given Inequalities

We are given two inequalities to consider: 1. \( x > 0 \) 2. \( x \geq -1 \) Both conditions must be satisfied simultaneously. This means we need the overlap or intersection of the solutions of these inequalities.

Step 2: Solve and Graph Each Inequality

First, let's represent each inequality on separate number lines:- For \( x > 0 \), draw a number line with an open circle (indicating non-inclusion) at 0 and shading to the right, covering all numbers greater than 0.- For \( x \geq -1 \), draw a number line with a closed circle (indicating inclusion) at -1 and shading to the right, covering all numbers greater or equal to -1.

Step 3: Find the Intersection

To find the solution that satisfies both inequalities, look for the overlap on the number line:- The graph of \( x \geq -1 \) starts at -1 and goes to positive infinity.- The graph of \( x > 0 \) starts just past 0 and goes to positive infinity.The region where both solutions overlap is from 0 (not including 0 itself) to positive infinity.

Step 4: Write the Interval Notation

The interval notation for the overlap we found is on the interval starting just after 0 to infinity. Since 0 is not included (because it's \( x > 0 \), not \( x \geq 0 \)), we use a parenthesis: \((0, \infty)\)

Key Concepts

Graphing Inequalities

Graphing inequalities helps us visually understand the range of possible solutions for an inequality. When graphing, each inequality has its own set of rules based on whether values are included or not in the solution set.

Here’s a simple guide to graph different inequalities on a number line:

• For strict inequalities like \(x > 0\), use an open circle. This indicates that 0 is not part of the solution.
• For inclusive inequalities like \(x \geq -1\), use a closed circle. This means that -1 is included in the solution.

Once you’ve marked the critical points with circles, shade the number line in the direction of possible solutions.

By graphing both \(x > 0\) and \(x \geq -1\), it becomes clear where both inequalities meet. This overlap is the solution to the combined inequalities.

Number Line

A number line is a simple tool that can be used to visually represent numbers and inequalities. It is essentially a straight line with numbers placed at equal intervals along it.

Here’s how to effectively use a number line to represent inequalities:

• Identify key points where the inequality changes its status—like 0 and -1 in our exercise.
• Use open circles for values not included (e.g., \(x > 0\)).
• Use closed circles for values included (e.g., \(x \geq -1\)).
• Shade the area where the inequality holds true, to show all the numbers that satisfy it.

This visual representation allows you to easily spot the solution set by seeing where the shaded areas overlap.

Interval Notation

Interval notation is a concise way of expressing the set of solutions for an inequality. It tells us the beginning and end of the solution range and whether or not these endpoints are included.

Key symbols for interval notation include:

• Parentheses \(()\) for numbers that are not included in the solution (open intervals).
• Brackets \([])\) for numbers that are included in the solution (closed intervals).

In the given exercise, the interval notation \((0, \infty)\) means that any number greater than 0 is part of the solution. This excludes 0 itself but includes all numbers up to infinity.

Solution Sets

Solution sets include all possible values that satisfy one or more inequalities. When working with multiple inequalities, it’s important to find where these solution sets overlap to determine the common values that satisfy all conditions.

For the inequalities \(x > 0\) and \(x \geq -1\) from the exercise, the solution set must satisfy both conditions simultaneously.

Steps to find a solution set include:

• Graph each inequality on a number line to find the individual solution sets.
• Look for the intersection, where both solution sets overlap, as this forms the common solution.
• Express this overlap using interval notation to clearly communicate the range of valid solutions.

In this exercise, the solution set is \((0, \infty)\), indicating that all numbers greater than 0 satisfy both conditions.