Problem 47 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(2.2 x<-19.8\) and \(-4 x<40\)

Step-by-Step Solution

Answer

The solution is \((-10, -9)\).

Step 1: Isolate the variable in the first inequality

We start by solving the first inequality, \(2.2x < -19.8\). To isolate \(x\), divide both sides by 2.2: \[ x < \frac{-19.8}{2.2} \] Calculating the right side gives: \[ x < -9 \]

Step 2: Isolate the variable in the second inequality

Next, solve the second inequality, \(-4x < 40\). Divide both sides by \(-4\), remembering to reverse the inequality sign because we're dividing by a negative number: \[ x > \frac{40}{-4} \] Calculating the right side gives: \[ x > -10 \]

Step 3: Combine the solutions of both inequalities

Since we have 'and' between the inequalities, we need the values of \(x\) that satisfy both \(x < -9\) and \(x > -10\). The overlapping range is \[ -10 < x < -9 \]

Step 4: Write the solution in interval notation

The solution set where both conditions are satisfied is from \(-10\) to \(-9\), not including \(-10\) or \(-9\). So the interval notation is: \((-10, -9)\)

Step 5: Graph the solution set

To graph \((-10, -9)\), draw a number line. Use open circles at \(-10\) and \(-9\) to indicate these points are not included, and shade the line segment between them. This visually represents all values of \(x\) that satisfy both inequalities.

Key Concepts

Interval notation

Interval notation is a concise way of representing a range of numbers. It's particularly useful when working with solutions to inequalities. The key components are the parentheses or brackets:

• Parentheses, like \((-10, -9)\), indicate that the endpoints are not included in the set. This is also known as an "open interval."
• Brackets, on the other hand, like \([-10, 9]\), would indicate that the endpoints are included, which is referred to as a "closed interval."

In our exercise, for the inequality \-10 < x < -9\, since the values at the endpoints are not part of the solution, we use parentheses for both -10 and -9. Thus, it is written as \((-10, -9)\).
Understanding and using interval notation correctly helps in communicating solutions clearly and efficiently.

Graphical representation

Graphical representation involves plotting the solution of inequalities on a number line. This visual representation helps in understanding the solution better by showing all possible values that satisfy the inequality. To graph the solution of \((-10, -9)\), you follow these steps:

• Draw a horizontal line representing the number line.
• Mark the points \(-10\) and \(-9\) with open circles. Open circles mean these points are not included in the solution set.
• Shade the segment between \(-10\) and \(-9\) to illustrate that this is the range of values satisfying the compound inequality.

This method not only aids in understanding but is essential for portraying inequalities succinctly and graphically.

Solving inequalities

Solving inequalities involves finding all values of a variable that make an inequality true. Similar to solving equations, but with some key differences, especially when multiplying or dividing by negative numbers. Let's look at the process:1. **Isolate the variable**: Like solving equations, the goal is to get the variable on one side.

• Example: For \(2.2x < -19.8\), divide both sides by 2.2 to find the values of \(x\).

2. **Reverse the inequality**: When dividing or multiplying an inequality by a negative number, the inequality sign must be flipped.

• Example: For \(-4x < 40\), divide by -4 to isolate \(x\), resulting in the inequality sign changing direction to \(x > -10\).

3. **Combine solutions for compound inequalities**: If the compound inequality uses "and," find the intersection of the solution sets.

• Example: Combine \(x < -9\) and \(x > -10\) to find where they overlap: \(-10 < x < -9\).

Solving compound inequalities is a fundamental skill that allows one to deal with real-world scenarios represented by inequalities.