Problem 50 - Algebra A Combined Function

Question

Solve each inequality. Write the solution set in interval notation. $$ (4 x-9)(2 x+5)<0 $$

Step-by-Step Solution

Answer

\((-\frac{5}{2}, \frac{9}{4})\)

Step 1: Find the Critical Points

To solve the inequality \((4x - 9)(2x + 5) < 0\), start by setting each factor equal to zero to find the critical points:First factor: \(4x - 9 = 0\)\[4x = 9\]\[x = \frac{9}{4}\]Second factor: \(2x + 5 = 0\)\[2x = -5\]\[x = -\frac{5}{2}\]The critical points are \(x = \frac{9}{4}\) and \(x = -\frac{5}{2}\).

Step 2: Set Up the Number Line

Mark the critical points \(x = -\frac{5}{2}\) and \(x = \frac{9}{4}\) on a number line. These points divide the number line into three intervals: \((-\infty, -\frac{5}{2})\), \((-\frac{5}{2}, \frac{9}{4})\), and \((\frac{9}{4}, \infty)\).

Step 3: Test the Intervals

Choose a test point from each interval to determine where the product \((4x - 9)(2x + 5)\) is negative:- For the interval \((-\infty, -\frac{5}{2})\), choose \(x = -3\): \((4(-3) - 9)(2(-3) + 5) = (-12 - 9)(-6 + 5) = (-21)(-1) = 21\) Sign: Positive- For the interval \((-\frac{5}{2}, \frac{9}{4})\), choose \(x = 0\): \((4(0) - 9)(2(0) + 5) = (-9)(5) = -45\) Sign: Negative- For the interval \((\frac{9}{4}, \infty)\), choose \(x = 3\): \((4(3) - 9)(2(3) + 5) = (12 - 9)(6 + 5) = (3)(11) = 33\) Sign: Positive

Step 4: Write the Solution in Interval Notation

The inequality \((4x - 9)(2x + 5) < 0\) holds true where the product is negative, which is the interval \((-\frac{5}{2}, \frac{9}{4})\).In interval notation, the solution is: \((-\frac{5}{2}, \frac{9}{4})\).

Key Concepts

Critical Points

When solving inequalities involving a product, the first step is to identify the critical points. Critical points occur where each factor of the inequality equals zero. For our inequality \((4x - 9)(2x + 5) < 0\), we set each factor equal to zero to find the turning points.

• First factor is \(4x - 9 = 0\): To solve, add 9 to both sides and divide by 4, which gives \(x = \frac{9}{4}\).
• Second factor is \(2x + 5 = 0\): To solve, subtract 5 and divide by 2, arriving at \(x = -\frac{5}{2}\).

These critical points, \(x = \frac{9}{4}\) and \(x = -\frac{5}{2}\), separate the number line into distinct intervals where the sign of the inequality can change.

Number Line Method

The number line method is a visual approach to solve inequalities. After finding the critical points, we use the number line to break the entire number system into regions. Mark your critical points on a number line, and you are left with intervals between and beyond these points.

• Point at \(x = -\frac{5}{2}\)
• Point at \(x = \frac{9}{4}\)

These points divide the number line into three regions:

• Region 1: \((\infty, -\frac{5}{2})\)
• Region 2: \((-\frac{5}{2}, \frac{9}{4})\)
• Region 3: \((\frac{9}{4}, \infty)\)

Investigating each section will help identify where the inequality is satisfied.

Interval Notation

Interval notation is a mathematical way of expressing a set of numbers within a range. After determining in which intervals the inequality holds true, express the solution using this notation. For instance, the solution \((4x - 9)(2x + 5) < 0\) is valid in the negative sign interval which is \((-\frac{5}{2}, \frac{9}{4})\).

• \((\) or \()\): Indicate that the endpoint is not included.
• \([\) or \(]\): Indicate that the endpoint is included.

In our case, the critical points themselves are not included since the inequality is strictly "less than," so parentheses are used around both \(-\frac{5}{2}\) and \(\frac{9}{4}\). Using interval notation provides a concise way of expressing the solution to an inequality.

Test Intervals

Testing intervals helps determine where on the number line the inequality holds. Choose points (test points) from each interval created by the critical points. Then, substitute these into the product \((4x - 9)(2x + 5)\) to check where the product is negative, as this shows where the inequality \(< 0\) is satisfied.

• Interval \((\infty, -\frac{5}{2})\): Test \(x = -3\), gives a positive product, so it doesn’t satisfy the inequality.
• Interval \((-\frac{5}{2}, \frac{9}{4})\): Test \(x = 0\), gives a negative product, indicating this region satisfies the inequality.
• Interval \((\frac{9}{4}, \infty)\): Test \(x = 3\), gives a positive product again, not satisfying the inequality.

Testing intervals provides a methodical way of verifying which parts of the number line satisfy your inequality, helping ensure accurate solutions.