Problem 26 - Elementary and Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ x \leq-4 \text { and } x \geq-7 $$

Step-by-Step Solution

Answer

The solution is \([-7, -4]\) and can be graphed as a closed interval from -7 to -4 on the number line.

Step 1: Understand the Inequality

The given problem is a compound inequality combining two inequalities: \(x \leq -4\) and \(x \geq -7\). These conditions must be satisfied simultaneously.

Step 2: Solve the Compound Inequality

Combine the inequalities by considering the overlap where both conditions are true. The value of \(x\) must be greater than or equal to \(-7\) and less than or equal to \(-4\). Thus, the inequality simplifies to \(-7 \leq x \leq -4\).

Step 3: Graph the Solution Set

To graph the solution set, draw a number line. Mark closed circles at \(-7\) and \(-4\) since these values are included (the inequality signs are \(\leq\) and \(\geq\)), and shade the region between them. This represents all numbers between \(-7\) and \(-4\), including \(-7\) and \(-4\).

Step 4: Write in Interval Notation

Since the solution set includes both endpoint values, use square brackets in interval notation. The solution in interval notation is \([-7, -4]\).

Key Concepts

Inequality Solution

A compound inequality consists of two or more simple inequalities that are joined by the words "and" or "or.” In our exercise, we are dealing with an "and" compound inequality, meaning that the solution must satisfy both individual inequalities at the same time. To solve a compound inequality like the one given, where you have \( x \leq -4 \) and \( x \geq -7 \): - Start by focusing on the logical intersection, which means finding where both conditions are true. - The overlap indicates that the solution set is the range of values which are greater than or equal to -7 and less than or equal to -4.Therefore, the compound inequality \( x \leq -4 \) and \( x \geq -7 \) simplifies to \( -7 \leq x \leq -4 \). This means any number between -7 and -4, including -7 and -4, will solve the inequality.

Interval Notation

Once you've found your solution in terms of an inequality, it's often beneficial to express it in interval notation. This notation is a shorthand way of depicting a range of numbers on the number line. For the compound inequality \( -7 \leq x \leq -4 \), the range of acceptable values includes the endpoints -7 and -4. - When the endpoints are included, as indicated by "less than or equal to" (\( \leq \)) or "greater than or equal to" (\( \geq \)), use square brackets [ ].- So, the interval notation for this solution is written as \([-7, -4]\).Interval notation offers a clear and efficient method of representing a solution set, making it particularly useful for expressing concepts quickly and precisely in mathematics.

Graphing Inequalities

Graphing inequalities provides a visual representation of the solution set on a number line, making it easier to understand which values satisfy the inequality.To graph the solution \( -7 \leq x \leq -4 \):- Draw a horizontal number line with markers that represent key numbers, in this case, -7 and -4. - At -7 and -4, draw closed circles to indicate that these numbers are included in the solution. The closed circle signifies that the inequality is \( \leq \) or \( \geq \), including the endpoint.- Shade the region between -7 and -4 to show all numbers between these endpoints are part of the solution.This graph illustrates that every point from -7 through -4, including the endpoints, are solutions to the compound inequality. Graphing inequalities helps in comprehending and communicating the range of solutions effectively.