Problem 5 - Algebra for College Students

Question

Solve each inequality and graph its solution set on a number line. $$(2 x-1)(3 x+7) \geq 0$$

Step-by-Step Solution

Answer

The solution set is \(x \in [-\frac{7}{3}, \frac{1}{2}] \cup [\frac{1}{2}, \infty)\).

Step 1: Identify critical points

To solve the inequality \((2x-1)(3x+7) \geq 0\), first determine where \((2x-1)(3x+7) = 0\). Solve \(2x-1=0\) to get \(x=\frac{1}{2}\) and solve \(3x+7=0\) to get \(x=-\frac{7}{3}\). These points divide the number line into intervals.

Step 2: Determine test intervals

The critical points are \(x = \frac{1}{2}\) and \(x = -\frac{7}{3}\). This divides the number line into three intervals: \((-\infty, -\frac{7}{3})\), \((-\frac{7}{3}, \frac{1}{2})\), and \((\frac{1}{2}, \infty)\).

Step 3: Test the sign in each interval

Select a test point from each interval to determine the sign of the expression in that interval. Choose \(x = -3\) for \((-\infty, -\frac{7}{3})\), \(x = 0\) for \((-\frac{7}{3}, \frac{1}{2})\), and \(x = 1\) for \((\frac{1}{2}, \infty)\).

Step 4: Evaluate the expression

Substitute the test points into \((2x-1)(3x+7)\):- For \(x = -3\), \((2(-3)-1)(3(-3)+7) = (-7)(-2) = 14 > 0\).- For \(x = 0\), \((2(0)-1)(3(0)+7) = (-1)(7) = -7 < 0\).- For \(x = 1\), \((2(1)-1)(3(1)+7) = (1)(10) = 10 > 0\).Thus, the intervals \((-\infty, -\frac{7}{3})\) and \((\frac{1}{2}, \infty)\) make the expression positive.

Step 5: Analyze the critical points

Since the inequality is \((2x-1)(3x+7) \geq 0\), include points where it equals zero. At \(x = -\frac{7}{3}\) and \(x = \frac{1}{2}\), the expression equals zero, so include these points as they satisfy the inequality.

Step 6: Write the solution set

Combine the results from the test intervals and critical point analysis. The solution set is \[x \in [-\frac{7}{3}, \frac{1}{2}] \, \cup \, [\frac{1}{2}, \infty)\].

Step 7: Graph the solution set

On a number line, draw filled circles at \(x = -\frac{7}{3}\) and \(x = \frac{1}{2}\) because these points are included. Shade the intervals \((-\infty, -\frac{7}{3}]\) and \([\frac{1}{2}, \infty)\) to indicate the solution set.

Key Concepts

Critical Points

In the realm of inequalities, critical points play a crucial role in dividing the number line into sections where the sign of the expression might change. By solving

• \( (2x-1) = 0 \), we obtain the critical point \( x = \frac{1}{2} \).
• Similarly, solving \( (3x+7) = 0 \) gives us another critical point at \( x = -\frac{7}{3} \).

These points do more than just mark changes on a number line—they help us determine where the expression equals zero and can indicate changes in the inequality's direction. These points are essential in shaping how we analyze the overall inequality.
To confirm each critical point, substitute back into the factors, ensuring they equal zero. This verification ensures correctness in later steps.

Test Intervals

Once you've identified the critical points, the next step in solving compound inequalities is to determine test intervals. These intervals are bounded by the critical points themselves and potentially extend into infinity on either side.
For our inequality,

• The intervals are: \((-\infty, -\frac{7}{3}) \),
• \((-\frac{7}{3}, \frac{1}{2}) \),
• \((\frac{1}{2}, \infty) \).

Each of these intervals must be "tested" by selecting a test point from within to determine the sign of the inequality in that section. By doing so, we find where the expression is zero, positive, or negative.
Choosing strategic test points within each interval allows us to effectively analyze the behavior of the inequality across the entire number line.

Solution Set

The solution set is the range of values that satisfy the given inequality. It's essentially your answer expressed in interval notation, taking into account all relevant critical points and test intervals.
For \((2x-1)(3x+7) \geq 0\):

• The intervals \((-\infty, -\frac{7}{3}] \) and
• \([\frac{1}{2}, \infty)\) indicate where the expression results in zero or positive values.

Therefore, combining these intervals, we derive the solution set \( x \in [-\frac{7}{3}, \frac{1}{2}] \, \cup \, [\frac{1}{2}, \infty) \).
This expression tells us that all numbers within these bounds, including the points \(-\frac{7}{3}\) and \(\frac{1}{2}\), satisfy our inequality.

Graphing Inequalities

Graphing inequalities helps visually represent the solution set on a number line. This graphical method can provide a clearer understanding and quick verification of the solution.
Here's how to graph the solution set:

• Draw a number line.
• Mark the critical points \(-\frac{7}{3}\) and \(\frac{1}{2}\) with filled circles, indicating these values are included in the solution.
• Shade the intervals \((-\infty, -\frac{7}{3}] \) and
• \([\frac{1}{2}, \infty)\), showing where the inequality holds.

This shading and highlighting technique effectively sets limits on the range of valid solutions and visually confirms the relationship between these solutions and the inequality. Understanding graphing as part of solving inequalities is useful for comprehending solution sets and inequalities on a broader scale.