Problem 33 - Algebra A Combined Function

Question

Solve each inequality. Write the solution set in interval notation. $$ (2 x-8)(x+4)(x-6) \leq 0 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -4] \cup [4, 6]\).

Step 1: Find the critical points

To solve the inequality \((2x - 8)(x + 4)(x - 6) \leq 0\), first find the values of \(x\) that make each factor zero. Set each factor equal to zero: 1. \(2x - 8 = 0\) leads to \(x = 4\). 2. \(x + 4 = 0\) leads to \(x = -4\). 3. \(x - 6 = 0\) leads to \(x = 6\). So the critical points are \(x = -4\), \(x = 4\), and \(x = 6\).

Step 2: Determine test intervals

The critical points \(-4\), \(4\), and \(6\) divide the number line into four intervals: 1. \((-\infty, -4)\)2. \((-4, 4)\)3. \((4, 6)\)4. \((6, \infty)\).

Step 3: Test intervals for inequality

For each interval, choose a test point and substitute it into the inequality to check the sign. 1. For \((-\infty, -4)\), use \(x = -5\): \((2(-5) - 8)(-5 + 4)(-5 - 6) = (-18)(-1)(-11) = -198\). Since -198 is negative, this interval satisfies the inequality. 2. For \((-4, 4)\), use \(x = 0\): \((2(0) - 8)(0 + 4)(0 - 6) = (-8)(4)(-6) = 192\). Since 192 is positive, this interval does not satisfy the inequality. 3. For \((4, 6)\), use \(x = 5\): \((2(5) - 8)(5 + 4)(5 - 6) = (2)(9)(-1) = -18\). Since -18 is negative, this interval satisfies the inequality. 4. For \((6, \infty)\), use \(x = 7\): \((2(7) - 8)(7 + 4)(7 - 6) = (6)(11)(1) = 66\). Since 66 is positive, this interval does not satisfy the inequality.

Step 4: Include critical points where the inequality equals zero

Check the inequality at critical points \(-4\), \(4\), and \(6\) since \(\leq 0\) includes equality.1. \((2(-4) - 8)(-4 + 4)(-4 - 6) = 0\), so \(x = -4\) is included. 2. \((2(4) - 8)(4 + 4)(4 - 6) = 0\), so \(x = 4\) is included.3. \((2(6) - 8)(6 + 4)(6 - 6) = 0\), so \(x = 6\) is included.

Step 5: Write final solution in interval notation

Combine the solution intervals \((-\infty, -4]\), \([4, 6]\) to write the solution set. The solution in interval notation is:\[ (-\infty, -4] \cup [4, 6] \]

Key Concepts

Critical Points

When solving polynomial inequalities, the first step is to find the critical points. These are the points where the polynomial equals zero. For our inequality \((2x - 8)(x + 4)(x - 6) \leq 0\), we'll solve each factor equated to zero:

• \(2x - 8 = 0\) implies \(x = 4\)
• \(x + 4 = 0\) implies \(x = -4\)
• \(x - 6 = 0\) implies \(x = 6\)

These values, \(x = -4\), \(x = 4\), and \(x = 6\), are the critical points. They divide the x-axis into intervals, and they indicate where the sign of the polynomial might change.

Interval Notation

After finding the critical points, it's important to express the solution of the inequality using interval notation. Interval notation provides a clean and concise way to represent the set of solutions. Let's look at how to use it.
First, determine which intervals to include based on the inequality. For \((2x - 8)(x + 4)(x - 6) \leq 0\), we need intervals that make the expression less than or equal to zero.
The critical points \(-4, 4,\) and \(6\) divide the real number line into different intervals. Interval notation uses round brackets \(()\) for excluding endpoints and square brackets \([]\) for including them when an inequality includes equality \(\leq\) or \(\geq\). You should remember:

• Use \([-4, -∞)\) and \((4, 6]\) for satisfying \((-)\) and equality.
• Combine these to form the solution in interval notation.

Test Intervals

Once you have your intervals, the critical task is to test these intervals to see where the inequality holds true. Choose a number (test point) from within each interval:

• \((-\infty, -4)\) test: \(x = -5\)
• \((-4, 4)\) test: \(x = 0\)
• \((4, 6)\) test: \(x = 5\)
• \((6, \infty)\) test: \(x = 7\)

For each test point, substitute it back into the inequality and solve:- Check if the result is positive or negative.- A negative product implies the inequality is satisfied in that section.- Consider both negative and zero results since the inequality is \(\leq 0\). This will guide which intervals are part of your solution.

Inequality Solution

The solution of an inequality involves putting together the tested intervals and considering the critical points separately. In this context, there are two main parts:

• Intervals that satisfy the inequality: These are the intervals for which the test points result in a negative value. For \((2x-8)(x+4)(x-6) \leq 0\), these intervals are \((-\infty, -4]\) and \([4, 6]\).
• Including critical points: Since this is a \(\leq\) inequality, we need to include the critical points where the expression equals zero: \(x = -4\), \(x = 4\), and \(x = 6\)

Once determined, combine these in interval notation. Here, the solution is:\((-\infty, -4] \cup [4, 6]\). This notation captures where the inequality is true on the number line.