Problem 47 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Solve the first inequality

We are given the inequality \(2.2x < -19.8\). To solve for \(x\), divide both sides by 2.2: \[ x < \frac{-19.8}{2.2}\]Calculating the right side gives us:\[ x < -9\]

Step 2: Solve the second inequality

We need to solve \(-4x < 40\). Divide both sides by -4, but remember to reverse the inequality sign because we are dividing by a negative number:\[x > \frac{40}{-4}\]This simplifies to:\[x > -10\]

Step 3: Combine the solutions

The compound inequality is \(x < -9\) and \(x > -10\). To find a solution that satisfies both conditions, the value of \(x\) must be both less than -9 and greater than -10. This gives us a narrow range:\[-10 < x < -9\]

Step 4: Express the solution in interval notation

Express the solution \(-10 < x < -9\) in interval notation. Since both -10 and -9 are not included in the solution, we use open brackets:\((-10, -9)\)

Step 5: Graph the solution set

On a number line, draw an open circle at -10 and another at -9. Shade the region strictly between -10 and -9, indicating all numbers between these two but not including them.

Key Concepts

Solving Inequalities

When it comes to solving inequalities, the process is quite similar to solving equations with a few additional rules to consider. Here, our task is to isolate the variable, often represented as \(x\), on one side of the inequality sign. There are a few key things to keep in mind:

• If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains the same. For example, dividing \(2.2x < -19.8\) by 2.2 gives \(x < -9\).
• However, if you multiply or divide by a negative number, the direction of the inequality sign must flip. For instance, dividing \(-4x < 40\) by -4 requires flipping the inequality from \(<\) to \(>\), resulting in \(x > -10\).

By following these rules carefully, we can solve each part of a compound inequality. The solutions then need to be considered together to find a range of values that satisfy both parts.

Graphing Solutions

Graphing solutions of inequalities involves visually depicting the range of values that satisfy the inequality. For a compound inequality like \(-10 < x < -9\), graphing on a number line can be quite helpful.

• Identify each critical point, which in this case are -9 and -10. In the graph, these are the points where the inequality changes from true to false.
• Since neither endpoint is included, we use open circles to indicate that -10 and -9 are not part of the solution.
• Shade the region between these open circles to show all the possible values for \(x\) that make the inequality true. This area between -10 and -9 represents the solution set.

Graphing gives a visual confirmation of our solution and helps in understanding the relationship between the values.

Interval Notation

Interval notation is a efficient way to express the solution sets of inequalities. It indicates which values are included in or excluded from the solution set. Here's how it's used:

• The format is \((a, b)\) for open intervals where \(a\) and \(b\) are not included, and \([a, b]\) for closed intervals where both \(a\) and \(b\) are included.
• In our example \(-10 < x < -9\), we use \((-10, -9)\) to show that the values between -10 and -9 are included in the solution, but -10 and -9 themselves are not.
• Interval notation is a compact and precise way to convey the range of solutions, making it particularly useful in mathematics.

Applying interval notation helps in simplifying the representation of a solution, especially when expressing it visually or in written form.