Problem 27 - Intermediate Algebra

Question

Solve each inequality. $$ (x+1)(x-3)^{2}>0 $$

Step-by-Step Solution

Answer

The solution is \((-1, 3) \cup (3, \infty)\).

Step 1: Identify Critical Points

To solve the inequality \((x+1)(x-3)^2 > 0\), first identify where each factor is equal to zero. Set each factor equal to zero: \(x+1=0\) gives \(x=-1\), and \((x-3)^2=0\) gives \(x=3\). These are the critical points, which divide the number line into intervals that we will test.

Step 2: Determine Intervals

The critical points \(x=-1\) and \(x=3\) divide the number line into three intervals: \((-\infty, -1)\), \((-1, 3)\), and \((3, \infty)\). We will test a point in each interval to see if the inequality holds.

Step 3: Test Interval \((-\infty, -1)\)

Choose a test point in the interval \((-\infty, -1)\), such as \(x=-2\). Substitute \(x=-2\) into \((x+1)(x-3)^2\): \((-2+1)(-2-3)^2 = (-1)(5)^2 = -25\). Since \(-25 < 0\), the inequality does not hold in this interval.

Step 4: Test Interval \((-1, 3)\)

Choose a test point in the interval \((-1, 3)\), such as \(x=0\). Substitute \(x=0\) into the expression: \((0+1)(0-3)^2 = 1(3)^2 = 9\). Since \(9 > 0\), the inequality holds in this interval.

Step 5: Test Interval \((3, \infty)\)

Choose a test point in the interval \((3, \infty)\), such as \(x=4\). Substitute \(x=4\) into the expression: \((4+1)(4-3)^2 = 5(1)^2 = 5\). Since \(5 > 0\), the inequality holds in this interval.

Step 6: Identify Solution Set

Since the inequality is satisfied in the intervals \((-1, 3)\) and \((3, \infty)\), we combine these intervals to write the solution set. Since the factor \((x-3)^2\) is non-negative and does not affect the inequality sign, \(x=3\) is an excluded boundary.

Key Concepts

Critical Points

In solving inequalities like \((x+1)(x-3)^2 > 0\), identifying critical points is a vital step. Critical points are values of \(x\) where the expression equals zero. In other words, they are where the sign of the expression could potentially change.

To find these critical points, we set each factor in the expression equal to zero:

• For \(x+1=0\), solving gives \(x=-1\).
• For \((x-3)^2=0\), solving gives \(x=3\).

These values, \(x=-1\) and \(x=3\), are crucial as they mark the potential points where the inequality changes its sign on a number line. Hence, they will help us define the intervals for the next steps.

Number Line Intervals

Once we have the critical points, we use them to divide the number line into distinct intervals where we will check the inequality.

• The critical point \(x=-1\) marks one boundary.
• The critical point \(x=3\) marks another.
  
  

This results in the formation of three separate number line intervals:

• \((-\infty,-1)\): To the left of \(x=-1\)
• \((-1,3)\): Between \(x=-1\) and \(x=3\)
• \((3,\infty)\): To the right of \(x=3\)

In each of these intervals, the expression could behave differently. Hence, testing within these intervals enables us to determine where the expression is positive, satisfying the original inequality.

Testing Intervals

With defined intervals, the next step is to test each one to determine where the inequality holds true. This involves selecting any easy-to-use test point from each interval and substituting it back into the original expression \((x+1)(x-3)^2\):

• In the interval \((-\infty,-1)\), a test point is \(x=-2\). Plugging this in gives a product of -25, which is less than zero, indicating the inequality does not hold.
• In the interval \((-1,3)\), a test point is \(x=0\). Here, substituting results in 9, a value greater than zero, showing that the inequality holds.
• In the interval \((3,\infty)\), using \(x=4\) as the test point yields 5, again greater than zero, confirming the inequality holds in this range as well.
  
  

Through these tests, we see that the inequality \((x+1)(x-3)^2>0\) is satisfied in the intervals \((-1, 3)\) and \((3, \infty)\). Combining these intervals without including critical point \(x=3\), which does not change the sign, provides the final solution set, ensuring a complete understanding of the inequality's behavior.