Problem 27 - Algebra A Combined Function

Question

Solve each inequality. Write the solution set in interval notation. $$ (2 x-3)(4 x+5) \leq 0 $$

Step-by-Step Solution

Answer

The solution set is \( \left[-\frac{5}{4}, \frac{3}{2}\right] \).

Step 1: Identify Critical Points

First, find the values of \( x \) where each factor of the inequality equals zero. For \( (2x-3) \), set it to zero: \[2x - 3 = 0 \Rightarrow x = \frac{3}{2}\]For \( (4x+5) \), set it to zero:\[4x + 5 = 0 \Rightarrow x = -\frac{5}{4}\]Now, the critical points are \( x = \frac{3}{2} \) and \( x = -\frac{5}{4} \).

Step 2: Test Intervals Between Critical Points

The critical points divide the number line into intervals: 1. \( (-\infty, -\frac{5}{4}) \)2. \( (-\frac{5}{4}, \frac{3}{2}) \)3. \( (\frac{3}{2}, \infty) \)Choose test points in each interval to determine where the inequality \((2x-3)(4x+5) \leq 0\) holds:- For \((-\infty, -\frac{5}{4})\), choose \( x = -2 \).- For \((-\frac{5}{4}, \frac{3}{2})\), choose \( x = 0 \).- For \((\frac{3}{2}, \infty)\), choose \( x = 2 \).

Step 3: Evaluate Test Points

Substitute each test point into the inequality to see if the expression is negative or zero:1. \((2(-2) - 3)(4(-2) + 5)\): \(( -4 - 3 )(-8 + 5) = -7 \times -3 = 21\) (positive)2. \((2(0) - 3)(4(0) + 5)\): \(( 0 - 3 )(0 + 5) = -3 \times 5 = -15\) (negative)3. \((2(2) - 3)(4(2) + 5)\): \(( 4 - 3 )(8 + 5) = 1 \times 13 = 13\) (positive)

Step 4: Write the Solution Set

Since the inequality is \( \leq 0 \), we choose intervals where the expression is non-positive. This only occurs in the interval where the test point \( x = 0 \) yielded a negative result. The solution interval is thus:\[ (-\frac{5}{4}, \frac{3}{2}) \]Include the critical points: \( x = -\frac{5}{4} \) and \( x = \frac{3}{2} \) because these values make one of the factors zero, resulting in zero. The final solution in interval notation is:\[ \left[-\frac{5}{4}, \frac{3}{2}\right] \]

Key Concepts

Interval Notation

Interval notation is a way of representing a set of numbers along a number line. It provides a shorthand method for expressing intervals. In interval notation:

• Brackets "] [" are used to include an endpoint, meaning the number is part of the interval.
• Parentheses "] (" are used to exclude an endpoint, meaning the number isn't part of the interval.

For example, the interval \([-2, 3)\) includes all the numbers from \(-2\) to fewer than \(3\). Using interval notation provides a compact and straightforward method to express the solution set of inequalities. This is particularly useful when describing where they hold true.

Critical Points in Inequalities

When solving inequalities, critical points are values that make the expression zero or undefined. Finding these points is essential because they help form the boundaries of intervals.

• For the inequality \( (2x-3)(4x+5) \leq 0 \), the critical points are found by setting each factor equal to zero.
• \(2x - 3 = 0\) gives \(x = \frac{3}{2}\), and \(4x + 5 = 0\) gives \(x = -\frac{5}{4}\)

These values split the number line into different sections or intervals. Identifying and understanding critical points helps us determine where the inequalities are satisfied.

Testing Intervals

After identifying critical points, the next step involves testing intervals. The intervals are the segments between and beyond the critical points.

• For \(x = -\frac{5}{4}\) and \(x = \frac{3}{2}\), the number line divides into three intervals: \((-\infty, -\frac{5}{4})\), \((-\frac{5}{4}, \frac{3}{2})\), and \( (\frac{3}{2}, \infty)\).
• Choose a test point inside each interval to see if the inequality holds. A test point is a single value from that interval that is easiest to work with.
• For example, you might choose \(x = -2\) for \((-\infty, -\frac{5}{4})\), \(x = 0\) for \((-\frac{5}{4}, \frac{3}{2})\), and \(x = 2\) for \( (\frac{3}{2}, \infty)\).

Testing these intervals helps identify where the inequality is negative or zero—ultimately leading us to the solution set.

Solution Sets for Inequalities

A solution set is simply all the possible values of \(x\) that satisfy the inequality. In our case, we are looking for where the inequality is non-positive ((2x-3)(4x+5) \leq 0). This means we look for intervals where the inequality evaluates to zero or a negative number.

• After testing, we find that the expression is negative in the interval \((-\frac{5}{4}, \frac{3}{2})\).
• Since the factors can be zero, points \(-\frac{5}{4}\) and \(\frac{3}{2}\) are included.
• Thus, the solution set is expressed in interval notation as \([-\frac{5}{4}, \frac{3}{2}]\).

Understanding solution sets allows students to grasp where and why certain values satisfy an inequality. It's a vital part of interpreting and solving real-life situations too.