Problem 27 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ -3 x>3 \text { and } x+3>0 $$

Step-by-Step Solution

Answer

(-3, -1)

Step 1 - Solve the first inequality

Start with the first inequality -3x > 3 Divide both sides by -3, and remember to reverse the inequality sign: x < -1

Step 2 - Solve the second inequality

Now solve the second inequality x + 3 > 0 Subtract 3 from both sides: x > -3

Step 3 - Determine the intersection

Identify the range where both inequalities are true. The first inequality is x < -1 The second inequality is x > -3 Intersection: -3 < x < -1 Now has been simplified to a simpler compound inequality.

Step 4 - Graph the solution set

Draw a number line. Mark open circles at -3 and -1, because these values are not included. Shade the region between -3 and -1.

Step 5 - Write the solution in interval notation

The solution set in interval notation is (-3, -1)

Key Concepts

solving inequalities

Inequalities describe a range of possible values, rather than just one specific value. When solving inequalities, the goal is to isolate the variable on one side. Begin by applying simple arithmetic operations. Be cautious: when you multiply or divide by a negative number, the inequality sign reverses. For instance, dividing both sides of \(-3x > 3\) by \(-3\) and flipping the inequality sign results in \(x < -1\). Look at each inequality individually to solve, then combine results.

interval notation

Interval notation provides a concise way to describe ranges of values. It uses brackets and parentheses to show whether endpoints are included or excluded. For our example, \(-3 < x < -1 \), both bounds (\(-3\) and \(-1\)) are not included. Therefore, the interval notation is \((-3, -1)\). When boundaries are included, use brackets, as in \([a, b]\). For unbounded intervals, use \(-\infty \) or \(+\infty\).

graphing inequalities

Graphing inequalities effectively visualizes solution sets. First, draw a number line. Identify critical points (such as \-3\ and \-1\ in our example). Use open circles for non-included values and closed circles for included ones. Shade the region representing the solution set—in this case, the area between \-3\ and \-1\, excluding both points, because the inequalities are \(-3 < x < -1\). This visually confirms the range of solutions.

number line

A number line graphically represents numbers in sequence. Use it to plot and compare numbers and solution sets. Mark specific points (such as -3 and -1). Use circles to indicate whether these points are included. Open circles denote values that are not included, while closed circles mean the value is part of the solution. Shade the regions between the appropriate points. For our inequality \(-3 < x < -1\), this means shading between \-3\ and \-1\, with open circles at both points.