Problem 103 - A Graphical Approach to College Algebra

Question

Solve each three-part inequality analytically. Support your answer graphically. $$4 \leq 2 x+2 \leq 10$$

Step-by-Step Solution

Answer

The solution to the inequality is \(1 \leq x \leq 4\).

Step 1: Break Down the Compound Inequality

The three-part inequality \(4 \leq 2x + 2 \leq 10\) can be split into two separate inequalities: \(4 \leq 2x + 2\) and \(2x + 2 \leq 10\). We will solve each inequality individually.

Step 2: Solve the First Inequality \(4 \leq 2x + 2\)

Subtract 2 from both sides of the inequality: \(4 - 2 \leq 2x\) which simplifies to \(2 \leq 2x\). Next, divide both sides by 2: \(\frac{2}{2} \leq x\), so \(1 \leq x\).

Step 3: Solve the Second Inequality \(2x + 2 \leq 10\)

Subtract 2 from both sides of the inequality: \(2x + 2 - 2 \leq 10 - 2\), resulting in \(2x \leq 8\). Then, divide both sides by 2: \(x \leq 4\).

Step 4: Combine the Solutions

From Steps 2 and 3, we have the inequalities \(1 \leq x\) and \(x \leq 4\). Combine these to form the compound inequality \(1 \leq x \leq 4\). This sets the range for the possible values of \(x\).

Step 5: Graphical Representation

Graph the solution set on a number line. Draw a solid line or bold segment from \(x = 1\) to \(x = 4\) and place closed circles at both ends to indicate that 1 and 4 are included in the solution set.

Key Concepts

Inequality Solving

When we talk about inequality solving, we are focusing on finding the set of numbers that satisfy the given inequalities. An inequality symbol, such as \( \leq \), \( \geq \), \( < \), or \( > \), tells us about the relationship between two expressions. In our three-part inequality \( 4 \leq 2x+2 \leq 10 \), we can observe two inequalities combined into one expression.

The initial step in solving this problem is to break the three-part inequality into two separate inequalities:

• \( 4 \leq 2x + 2 \)
• \( 2x + 2 \leq 10 \)

These inequalities essentially limit our variable \( x \) to a certain range, which we then solve to find this range. The goal is to isolate \( x \) and find all possible values that satisfy these two conditions. Solving each inequality separately allows us to understand the individual constraints that \( x \) must respect.

Graphical Representation

Graphical representation is a vital aspect when it comes to understanding inequalities visually. After solving the compound inequality \( 1 \leq x \leq 4 \), it’s crucial to graph these results to visualize the range of solutions. A number line is often used to sketch the solutions of inequalities, providing an intuitive way to see the values that \( x \) can take.

The graphical representation involves marking the boundary points with circles and drawing a line segment between them on the number line. In our example, closed circles on the number line at \( x = 1 \) and \( x = 4 \) indicate that these end points are included in the solution, as denoted by the 'less than or equal to' symbols. This helps bring clarity, showing exactly which numbers are part of the solution.

Compound Inequality

A compound inequality consists of two separate inequalities joined together, usually indicating a range of solutions. In our problem, the compound inequality \( 4 \leq 2x + 2 \leq 10 \) combines two conditions into one overarching statement.

Working with compound inequalities often requires you to split them into simpler, more manageable parts. Once split, each part can be solved individually. For instance, solving \( 4 \leq 2x + 2 \) and \( 2x + 2 \leq 10 \) separately allows us to find values of \( x \) that satisfy each inequality. The solutions are then combined, offering a clearer description of the feasible solutions. This process emphasizes the synergy between different inequality conditions and the need to consider them collectively.

Number Line Representation

The number line representation is a graphical technique used to display the solution set of inequalities. In our example, the number line is a simple tool showing the range from \( x = 1 \) to \( x = 4 \). Here's how to create this:

• Begin by drawing a horizontal line.
• Mark the important points, \( x = 1 \) and \( x = 4 \), on this line.
• Place closed circles on these points to signify they are part of the solution set.
• Shade or draw a solid line segment between these points to indicate that all numbers from 1 to 4 (inclusive) are solutions.

This representation not only reinforces what was solved numerically but also provides a visual confirmation of where \( x \) lies according to the problem. The use of closed circles and solid lines clarifies inclusion of endpoints, making the solution clear and accessible.