Problem 7 - Elementary Algebra Within Reach

Question

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} x<3 \\ x>-2 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system of inequalities is \(x > -2\) and \(x < 3\), meaning all the numbers between -2 and 3, exclusive, form the solution to the system.

Step 1: Understand the Inequalities

We have a system of two linear inequalities: \(x < 3\) and \(x > -2\). The inequality \(x < 3\) means that \(x\) is less than 3. Likewise, the inequality \(x > -2\) means that \(x\) is greater than -2. This system, therefore, restricts \(x\) to a certain range between -2 and 3.

Step 2: Sketch the Inequalities on a Number Line

To sketch these inequalities on a number line, first draw a straight line. This line will represent all possible values of \(x\). Mark the points -2 and 3 on the number line. For \(x > -2\), draw an open circle at -2, and draw a line extending to the right of -2. An open circle is used to indicate that -2 is not part of the solution for this inequality. For \(x < 3\), draw an open circle at 3, and draw a line extending to the left of 3. Again, an open circle is used to indicate that 3 is not part of the solution for this inequality.

Step 3: Interpret the Overlapping Region

You will notice that there is an overlapping region on the number line between -2 and 3. This overlapping region indicates the solution to the system of inequalities. In other words, any number between -2 and 3 (exclusive) satisfies both inequalities.

Key Concepts

Number Line Representation

Graphing linear inequalities often begins with representing them on a number line. When dealing with inequalities like \(x < a\) or \(x > b\), we use a number line to easily visualize the range of possible values. To represent an inequality on the number line:

• Draw a horizontal line, which will act as your number line.
• Mark critical points, such as the values involving the inequalities (in this case, -2 and 3).
• Use open circles to denote that these points are not included in the inequality—this is crucial for indicating values that are less than or greater than the boundary.

The open circles at -2 and 3 remind us that -2 and 3 themselves are not solutions to these inequalities. Watching how these markers and lines extend on the number line helps us envision where our solutions begin and end.

Solving Inequalities

Solving linear inequalities is similar to solving equations but with specific rules due to the inequality sign. When we solve, our aim is to determine which values satisfy each condition:

• For \(x < 3\), we are looking for numbers that are smaller than 3. This inequality ensures we include numbers from negative infinity up to, but not including, 3.
• For \(x > -2\), we are after numbers greater than -2, meaning any number from just above -2 to positive infinity.

This results in two separate regions on the number line. When solving multiple inequalities together, evaluate where these regions coincide to find the solution that satisfies all conditions.

Overlapping Region Interpretation

The last step in graphing linear inequalities is identifying where solutions overlap. This overlapping region portrays the range of values that satisfy both inequalities simultaneously. In our task, the overlap on the number line occurs between the numbers -2 and 3.

• This stretch is exclusive of the endpoints, as indicated by open circles.
• The overlapping segment is easy to spot and is the finite interval that covers the values both inequalities agree on. For our exercise, this spans from \(-2\) to \(3\) (not including -2 and 3 themselves).

Understanding this overlap helps ensure that you've found the common ground between all conditions, and thus, the valid solutions. This visual and conceptual method ties together the system of inequalities logically and intuitively.