Problem 69 - College Algebra

Question

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{x^{2}-x-2}{x^{2}-4 x+3}>0 $$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < -1\), \(-1 < x < 1\), and \(x > 3\).

Step 1: Factorize the polynomial

Simplify the given inequality by factorizing the polynomials in the numerator and the denominator.\n So, \[\frac{(x - 2)(x + 1)}{(x - 1)(x - 3)}>0\]

Step 2: Compute critical points

The critical points of the inequality are computed by setting the numerator and denominator equal to zero. Therefore the critical points are \(x = 2\), \(x = -1\), \(x = 1\), and \(x = 3\)

Step 3: Create intervals

Next, divide the number line into intervals using the critical points. The intervals are \(x < -1\), \(-1 < x < 1\), \(1 < x < 2\), \(2 < x < 3\), and \(x > 3\)

Step 4: Testing the intervals

Choose a test value from each interval and substitute it into the simplified inequality to determine if the resulting statement is true.For \(x < -1\), choose \(x = -2\). Substituting \(-2\) into the inequality results in \(\frac{12}{15}>0\), which is true.For \(-1 < x < 1\), choose \(x = 0\). The inequality becomes \(\frac{-2}{-3}>0\), which is also true.For \(1 < x < 2\), choose \(x = 1.5\). The inequality becomes \(\frac{1.75}{-0.5}>0\), which is false.For \(2 < x < 3\), choose \(x = 2.5\). The inequality becomes \(\frac{2.75}{-0.5}>0\), again also false.For \(x > 3\), choose \(x = 4\). The inequality becomes \(\frac{6}{1}>0\), which is true.

Step 5: Write down the solution and draw the number line

From the test results, the solution to the inequality is \(x < -1\), \(-1 < x < 1\), and \(x > 3\). So, draw the solution intervals on the number line, labeling the critical points and highlighting the solution intervals.

Key Concepts

Polynomial Factorization

Understanding how to break down a polynomial into its constituent factors is a crucial first step in solving rational inequalities. Polynomial factorization is an essential tool because it simplifies complex expressions, making it easier to identify critical points and analyze the behavior of the function.

When dealing with rational inequalities like \(\frac{x^{2}-x-2}{x^{2}-4x+3}>0\), we look for factor pairs that when multiplied, give us the original polynomial expressions for the numerator and the denominator. In our example, \(x^{2}-x-2\) factors into \((x-2)(x+1)\), and \(x^{2}-4x+3\) factors into \((x-1)(x-3)\). This process reveals the values of 'x' that satisfy the equality, which are also the potential boundary points of our inequality's solution set.

Critical Points in Inequalities

Determining the critical points, or boundaries, where the inequality could change from true to false is a vital step in analyzing rational inequalities. The critical points are typically the zeroes of the numerator and denominator after factorization. These are the x-values where the rational function's graph intersects or approaches the x-axis.

In our example \(\frac{(x-2)(x+1)}{(x-1)(x-3)}>0\), setting the numerator \((x-2)(x+1)\) and denominator \((x-1)(x-3)\) equal to zero gives us the critical points: \(x=2\), \(x=-1\), \(x=1\), and \(x=3\). These critical points divide the number line into intervals we will test to determine where the inequality holds true.

Interval Testing Method

Once we have our critical points, the interval testing method comes into play. This method helps us figure out which intervals on the number line satisfy the inequality. We select test points from each interval, plug them into the simplified inequality, and check if they make the inequality true or false.

For instance, using test points such as \(x=-2\), \(x=0\), \(x=1.5\), \(x=2.5\), and \(x=4\) for their respective intervals around the critical points, we can determine whether the intervals lie above or below the x-axis, indicating where the original inequality is satisfied. By confirming which intervals produce a true statement when substituted into the inequality, we learn exactly where the solution set lies.

Graphing on a Number Line

After using the interval testing method, graphing on a number line is your final visualization tool. A number line allows you to clearly mark and understand the solution set of the rational inequality. Using the critical points and the results from the interval testing, you can graphically illustrate the regions where the inequality holds true.

For the inequality presented, the solution is \(x < -1\), \(-1 < x < 1\), and \(x > 3\). By plotting these intervals on the number line and marking the critical points, we see a visual representation of our solution. Open dots are used to represent values that are not included in the solution set (due to non-inclusive inequality), while filled dots indicate included values. Lines or rays are then drawn to depict the ranges of numbers that solve the inequality.