Problem 36 - Beginning and Intermediate Algebra

Question

Solve each inequality, and graph the solution set. $$ (x+2)(4 x-3)(2 x+7) \geq 0 $$

Step-by-Step Solution

Answer

The solution set is \([-\frac{7}{2}, -2] \cup [\frac{3}{4}, \infty)\).

Step 1 - Find the Roots of the Inequality

Set the inequality expression to zero to find the roots:\( (x+2)(4x-3)(2x+7) = 0 \).Solve each factor:1. \( x + 2 = 0 \rightarrow x = -2 \)2. \( 4x - 3 = 0 \rightarrow x = \frac{3}{4} \)3. \( 2x + 7 = 0 \rightarrow x = -\frac{7}{2} \)The roots are: \( x = -2 \), \( x = \frac{3}{4} \), and \( x = -\frac{7}{2} \).

Step 2 - Determine Intervals

Using the roots, divide the number line into intervals to test within the inequality:1. \( (-\infty, -\frac{7}{2}) \)2. \( (-\frac{7}{2}, -2) \)3. \( (-2, \frac{3}{4}) \)4. \( (\frac{3}{4}, \infty) \)

Step 3 - Test Each Interval

Select a test point from each interval and substitute into the inequality to check for truth values:1. Test point: \( x = -4 \) in \( (-\infty, -\frac{7}{2}) \): \( (-4+2)(4(-4)-3)(2(-4)+7) = (-2)(-19)(-1) = -38 \rightarrow \, \text{false} \)2. Test point: \( x = -3 \) in \( (-\frac{7}{2}, -2) \): \( (-3+2)(4(-3)-3)(2(-3)+7) = (-1)(-15)(1) = 15 \rightarrow \, \text{true} \)3. Test point: \( x = 0 \) in \( (-2,\frac{3}{4}) \):\( (0+2)(4(0)-3)(2(0)+7) = (2)(-3)(7) = -42 \rightarrow \, \text{false} \)4. Test point: \( x = 1 \) in \( (\frac{3}{4}, \infty) \):\( (1+2)(4(1)-3)(2(1)+7) = (3)(1)(9) = 27 \rightarrow \, \text{true} \).

Step 4 - Identify the Solution Set

Combine the intervals where the inequality is true and include the roots because the inequality is \( \geq \) (non-strict):Hence the solution set is:\( [-\frac{7}{2}, -2] \cup [\frac{3}{4}, \infty) \).

Step 5 - Graph the Solution Set

On the number line, shade the intervals \( [-\frac{7}{2}, -2] \) and \( [\frac{3}{4}, \infty) \). Draw solid circles at \( x=-\frac{7}{2} \), \( x=-2 \), and \( x=\frac{3}{4} \) to show the values as part of the solution set.

Key Concepts

Algebraic Inequality

An algebraic inequality is a mathematical statement that shows the relationship between two expressions using inequality symbols such as: \( < \), \( \leq \), \( > \), and \( \geq \). Solving an algebraic inequality involves finding the values of the variable that make the inequality true.
In our example, we have (x+2)(4x-3)(2x+7) \geq 0. Here, we need to find when the expression is greater than or equal to zero.
We start by finding the roots of each polynomial factor, then test intervals between these roots to determine where the inequality holds.

Graphing Inequalities

Graphing inequalities on a number line helps visualize the solution set. To graph an inequality:

• Identify critical points (roots or boundary points).
• Determine if these points are part of the solution (based on \( \leq \) or \( \geq \)). If they are, use solid circles; if not, use open circles.
• Test the intervals divided by these points to see where the inequality holds true.

For \((x+2)(4x-3)(2x+7) \geq 0\):

• Identify the roots: \( x = -2 , \ x = \frac{3}{4} \), and \( x = -\frac{7}{2} \)
• Test intervals between these roots.
• Plot the intervals on the number line where the inequality is satisfied.

Intervals on a Number Line

An interval on a number line represents a range of values. There are different types of intervals, such as:

• Open intervals \((a, b)\) where endpoints are not included.
• Closed intervals \([a, b] \) where endpoints are included.
• Semi-open intervals like \([a, b)\) or \((a, b]\).

To solve our inequality:

• We divide the number line using roots: \(x = -2, x = \frac{3}{4}, x = -\frac{7}{2}\).
• Create the intervals: \( ( -\infty, -\frac{7}{2} ),\ ( -\frac{7}{2}, -2 ), \ ( -2, \frac{3}{4} ), \ ( \frac{3}{4}, \infty )\).
• Test a point from each interval to determine where the expression is non-negative (greater than or equal to zero).

The solution is the union of intervals where the test points prove true.

Roots of Polynomial Inequalities

Roots of polynomials are values where the polynomial is equal to zero. Finding these roots is crucial for solving inequalities. For polynomial inequalities:

• Set each factor to zero.
• Solve for the variable to get the roots.
• These roots divide the number line into intervals to test for the inequality.

In \((x+2)(4x-3)(2x+7) \geq 0\):

• We solve for roots: \( x = -2, x = \frac{3}{4}, x = -\frac{7}{2}\).
• These roots help split the number line into useful intervals for testing.

Understanding how roots segment the number line helps determine where the polynomial's value meets the inequality criteria.