Problem 20 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ x<5 \text { and } x>0 $$

Step-by-Step Solution

Answer

The solution set is (0, 5).

Step 1: Understand the compound inequality

The given compound inequality is composed of two individual inequalities connected by 'and'. These are: 1. x < 5 2. x > 0

Step 2: Solve each individual inequality

To solve each part of the compound inequality:1. For x < 5: Any value of x less than 5 satisfies this inequality.2. For x > 0: Any value of x greater than 0 satisfies this inequality.

Step 3: Combine the solutions of both inequalities using 'and'

Since the inequalities are combined with 'and', we find the intersection of the solution sets of x < 5 and x > 0. The combined solution is all values of x that are both less than 5 and greater than 0.Hence, the solution is 0 < x < 5.

Step 4: Write the solution in interval notation

The interval notation for the solution 0 < x < 5 is (0, 5). This notation specifies all real numbers between 0 and 5, not including 0 and 5 themselves.

Step 5: Graph the solution set

To graph the solution set on a number line, draw an open circle at x = 0 and another open circle at x = 5. Shade the region between 0 and 5, not including the endpoints.

Key Concepts

interval notation

Interval notation helps to describe the set of solutions in a clear and concise manner. It uses brackets and parentheses to show which numbers are included or not included in the set. For instance, in the inequality solution \(0 < x < 5\), the interval notation is written as \((0, 5)\). This tells us that 0 and 5 are not included in the solution set because we use parentheses.
If an inequality does include the endpoint values, we would use brackets instead. For example, if the inequality was \(0 \leq x \leq 5\), the interval notation would be \([0, 5]\), indicating 0 and 5 are part of the solution set.
Interval notation is a compact way to communicate which parts of the number line are solutions to an inequality or a compound inequality. To summarize:

• Parentheses, \(( \text{ and } )\), indicate that endpoints are not included.
• Brackets, \([ \text{ and } ])\), show that endpoints are included.

number line graphing

Graphing inequalities on a number line gives a visual representation of the solution set. Let's take the example compound inequality \(0 < x < 5\). We start by drawing a horizontal line, our number line, and mark points at 0 and 5.
To graph the inequality on the number line:

• Place an open circle at 0 and at 5, because these points are not included in the solution.
• Shade the region between 0 and 5 to represent all numbers that satisfy the inequality.

Open circles indicate that particular points are not part of the solution set, while shaded regions show the range of values that are included in the set. Practicing number line graphing helps in better understanding and interpreting inequalities.

inequality intersection

An inequality intersection occurs when you combine two or more inequalities and find the set of values that satisfy all given conditions simultaneously. In our example, we have two inequalities linked by 'and': \(x < 5\) and \(x > 0\).
To determine the intersection:

• Solve both inequalities separately first.
• Combine the solutions while looking for the overlapping values.

For \(x < 5\), any value less than 5 satisfies this condition. Similarly, for \(x > 0\), any value greater than 0 is a solution. The intersection is the set of all values that are both less than 5 and greater than 0, which is expressed as \(0 < x < 5\).
Intersection in mathematics helps in understanding how different conditions or constraints apply together. Knowing this is crucial for solving compound inequalities and reading their solution sets accurately. Always look for overlapping intervals or common values when dealing with intersections.