Problem 26 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \\([-7, -4]\\).

Step 1: Analyze the Compound Inequality

We begin by understanding the compound inequality provided: \(x \leq -4\) and \(x \geq -7\). This compound inequality suggests that we are looking for values of \(x\) that satisfy both \(x \leq -4\) and \(x \geq -7\).

Step 2: Determine the Overlapping Region

Next, combine the two inequalities to find an intersection where both conditions hold true. The values of \(x\) must be less than or equal to -4 **and** greater than or equal to -7. Therefore, the intersection of these inequalities gives us \(-7 \leq x \leq -4\).

Step 3: Represent the Solution on a Number Line

To graph the solution set, plot a number line and shade the region from -7 to -4, including the endpoints, since the inequalities are inclusive (use closed dots or brackets on the endpoints).

Step 4: Write the Solution in Interval Notation

Express the solution set in interval notation, taking into account that the endpoints are included (closed interval). The interval notation is \([-7, -4]\).

Key Concepts

Solution Set

A solution set is a group of values that fulfill a given inequality or system of inequalities. In our case, the compound inequality is composed of two parts:

• The first part is that \(x\) must be less than or equal to \(-4\), symbolically represented as \(x \leq -4\).
• The second part requires \(x\) to be greater than or equal to \(-7\), represented as \(x \geq -7\).

When both these conditions are satisfied simultaneously, we determine the intersection of these inequalities. The solution set for these conditions is the range of values satisfying both, which is \(-7 \leq x \leq -4\).
This means that any number within this range, including both endpoints \(-7\) and \(-4\), is part of the solution set.

Graphing Inequalities

Graphing inequalities involves visually representing the solution set on a number line. For compound inequalities like \(-7 \leq x \leq -4\), you need to:

• Draw a horizontal number line with appropriate scaling.
• Locate and mark the critical points \(-7\) and \(-4\) on this number line.
• Since the inequalities are inclusive (contain equal signs), indicate these points with closed dots or brackets to signify that these endpoints are part of the solution.
• Shade the entire segment of the number line between \(-7\) and \(-4\) to reflect all values of \(x\) that satisfy the compound inequality.

This shaded portion is a visual representation of the solution set.

Interval Notation

Interval notation is a succinct way to express the solution set of inequalities. It helps to efficiently communicate the range of values that satisfy an inequality. For our compound inequality \(-7 \leq x \leq -4\), interval notation comes in handy:

• Use brackets \([\) and \(]\) to include endpoints when the inequality is \(\leq\) or \(\geq\), because these signify that the boundary points are part of the solution.
• Write the smallest number of the range first, followed by the largest, separated by a comma. So here, it is \([-7, -4]\).

Interval notation is particularly useful as it is both compact and clear, providing an immediate understanding of the solution without needing to deconstruct more complex mathematical language.