Problem 55 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ x<-1 \text { and } x>-5 $$

Step-by-Step Solution

Answer

The solution is \((-5, -1)\).

Step 1: Understand the problem

The problem involves solving a compound inequality. The given inequalities are: 1. \(x < -1\)2. \(x > -5\)Since it uses 'and', we are looking for the values of \(x\) that satisfy both conditions simultaneously.

Step 2: Solve each inequality

Solve the inequalities separately: 1. For the first inequality, \(x < -1\).2. For the second inequality, \(x > -5\).We will combine these to find the values of \(x\) that satisfy both conditions.

Step 3: Combine the inequalities

Since both conditions must be satisfied, we need the intersection of the solutions. This will give: \(-5 < x < -1\).

Step 4: Write the solution in interval notation

The solution in interval notation is written as: \((-5, -1)\).

Step 5: Graph the solution

To graph the solution, plot an open circle at \(-5\) and \(-1\) on the number line and shade the region between them. The open circles represent that the endpoints are not included in the interval.

Key Concepts

Interval notation

Interval notation is a mathematical way to describe a set of numbers along a number line. It is compact and specifies the start and end points of the interval. In our problem, we solved the compound inequality to get the solution:

\(-5 < x < -1 \).

This means that x is greater than -5 but less than -1. In interval notation, we write this as: \( (-5, -1) \).

Here's how to interpret interval notation:

• Parentheses, \( () \), indicate that the endpoint is not included (also known as an open interval).
  
• Brackets, \( [] \), indicate that the endpoint is included (also known as a closed interval).
  

In this problem, both -5 and -1 are not included, hence we use parentheses.

Graphing solution set

Graphing the solution set is a visual way to represent the interval on a number line. For our solution, \(-5 < x < -1\), here's how you can graph it:

• Draw a horizontal number line.
  
• Locate and mark -5 and -1 on the number line.
  
• Place an open circle at -5 and -1 to represent that these points are not included in the interval.
  
• Shade the region between -5 and -1 to show all the numbers between these two values.
  

The open circles at -5 and -1 visually indicate that the endpoints are not part of the solution. The shaded region between these circles represents all the numbers that satisfy the given compound inequalities.

Intersection of inequalities

When dealing with compound inequalities using 'and', you are looking for values that satisfy both inequalities at the same time. This is known as finding the intersection of inequalities.

For the inequalities \( x < -1 \) and \( x > -5 \), we want the values of x where both conditions are true.

\- Combining these inequalities, we get: \(-5 < x < -1\).
\- This means we are looking for x-values that are greater than -5 and less than -1 simultaneously.

The solution to compound inequalities using 'and' is the overlap (or intersection) of the solutions to each individual inequality. The interval \( (-5, -1) \) represents this intersection and includes all x-values that make both inequalities true.