Problem 58 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(-4 \leq-2(x+8)<8\)

Step-by-Step Solution

Answer

Solution: \(-12 < x \leq -6\), Interval: \((-12, -6]\).

Step 1: Break Down the Compound Inequality

The given compound inequality is \(-4 \leq -2(x+8) < 8\). This means we have two separate inequalities to solve: 1) \(-4 \leq -2(x+8)\) 2) \(-2(x+8) < 8\). We will solve each inequality separately and then find the intersection of the solutions.

Step 2: Solve the First Inequality

Solve \(-4 \leq -2(x+8)\):1. Distribute the -2: \(-4 \leq -2x - 16\)2. Add 16 to both sides: \(12 \leq -2x\)3. Divide both sides by -2 (remember to reverse the inequality sign): \(-6 \geq x\), or equivalently, \(x \leq -6\).

Step 3: Solve the Second Inequality

Solve \(-2(x+8) < 8\):1. Distribute the -2: \(-2x - 16 < 8\)2. Add 16 to both sides: \(-2x < 24\)3. Divide both sides by -2 (remember to reverse the inequality sign): \(x > -12\).

Step 4: Find the Intersection of the Solutions

Now, we find the intersection of the solutions \(x \leq -6\) (from Step 2) and \(x > -12\) (from Step 3). The intersection is the set of values of \(x\) that satisfy both conditions.Therefore, \(-12 < x \leq -6\) is the solution to the compound inequality.

Step 5: Express the Solution in Interval Notation

Express the solution set \(-12 < x \leq -6\) using interval notation. Since \(xeq -12\) (not included) but \(x = -6\) (included):The interval notation is \((-12, -6]\).

Step 6: Graph the Solution Set

To graph the solution set \((-12, -6]\), draw a number line:1. Place an open circle at -12 to indicate that -12 is not included in the solution.2. Place a closed circle at -6 to indicate that -6 is included.3. Shade the region between these two points to represent all \(x\) values in the interval \((-12, -6]\).

Key Concepts

Inequality Solution

Inequality solutions involve finding values that make an inequality true. In this exercise, we dealt with a compound inequality \(-4 \leq -2(x+8) < 8\). To address it, we solved two separate inequalities:

• \(-4 \leq -2(x+8)\)
• \(-2(x+8) < 8\)

The goal was to find an intersection of the solutions to these inequalities. This means we identified the range of values that satisfy both conditions.
For the first inequality, we used algebraic manipulation: distributing, adding, and dividing. It's crucial to remember reversing the inequality sign when dividing by a negative number.
In the second inequality, similar processes were applied: distribute and solve to isolate \(x\). The solutions were combined by finding the common values between them. This is a classic approach used in solving compound inequalities.

Interval Notation

Interval notation is a mathematical tool used to describe a range of numbers concisely. It makes stating solutions quick and precise.In this example, the solution set \(-12 < x \leq -6\) was expressed as \((-12, -6]\).

• A parenthesis \(()\) is used to show that a number is not included in the interval, like \(-12\).
• A bracket \([]\) indicates that a number is included, like \(-6\).

This notation highlights the boundary conditions of inequality solutions.Expressing solutions in this form allows for a compact view of ranges. It's especially useful in graphical representations and further mathematical operations.It is simple and clear, showing both upper and lower limits at a glance.

Number Line Graphing

Graphing inequalities on a number line visually represents the set of possible solutions. In our case, the solution \((-12, -6]\) was depicted on a number line:
Steps to graph this were as follows:

• Position an open circle at \(-12\) to show it's not part of the solution.
• Place a closed circle at \(-6\), indicating it is included.
• Shading the area between \(-12\) and \(-6\) distinguishes all valid \(x\) values.

These visual cues help understand the solution's scope. Open and closed circles guide reading whether points are included or not. Graphing transforms the abstract concept of inequalities into something tangible and easier to grasp. This method is a standard way to illustrate intervals within mathematics.