Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ x^{2}-5 x-6>0 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -1) \cup (6, \infty)\).

Step 1: Identify Critical Points

To solve the inequality \(x^2 - 5x - 6 > 0\), we begin by finding the critical points where the expression equals zero. Set the quadratic equation equal to zero: \(x^2 - 5x - 6 = 0\).

Step 2: Solve Quadratic Equation

Factor the quadratic equation: \((x-6)(x+1) = 0\). This gives the solutions \(x = 6\) and \(x = -1\).

Step 3: Determine Intervals

The critical points divide the number line into intervals: \((-\infty, -1)\), \((-1, 6)\), and \((6, \infty)\). We will test these intervals to find where the inequality holds.

Step 4: Test Intervals

Choose a test point from each interval to determine if it satisfies the inequality \(x^2 - 5x - 6 > 0\):- For \((-\infty, -1)\), choose \(x = -2\): \((-2)^2 - 5(-2) - 6 = 4 + 10 - 6 = 8 > 0\).- For \((-1, 6)\), choose \(x = 0\): \(0^2 - 5(0) - 6 = -6 < 0\).- For \((6, \infty)\), choose \(x = 7\): \(7^2 - 5(7) - 6 = 49 - 35 - 6 = 8 > 0\).

Step 5: Write Interval Notation

Based on the test results, the inequality is satisfied in intervals \((-\infty, -1)\) and \((6, \infty)\). Hence, the solution set in interval notation is \((-\infty, -1) \cup (6, \infty)\).

Step 6: Graph the Inequality

On a number line, graph the intervals where the inequality is true. Mark open circles at \(x = -1\) and \(x = 6\) because the inequality is strict (greater than, not equal to). Shade the regions \((-\infty, -1)\) and \((6, \infty)\).

Key Concepts

Understanding Critical Points in Quadratic Inequalities

When solving quadratic inequalities, finding the critical points is a key step. Critical points are the values of \(x\) where the quadratic equation equals zero. These points help us determine the intervals on a number line to test for solutions. To find critical points for the inequality \(x^2 - 5x - 6 > 0\), set the equation to zero: \(x^2 - 5x - 6 = 0\).

Once you rewrite the equation as \((x-6)(x+1) = 0\), you'll find the critical points at \(x = 6\) and \(x = -1\). These points divide the number line into distinct intervals:

• \((\infty, -1)\)
• \((-1, 6)\)
• \((6, \infty)\)

Each interval can then be tested to determine where the original inequality holds true.

Mastering Interval Notation

Interval notation is a system of writing the set of solutions to an inequality and presents them in a concise format. This method uses parentheses \(()\) and brackets \([]\) to describe which parts of the number line are included in the solution set.

For the inequality \(x^2 - 5x - 6 > 0\), we found that the solution exists in the intervals \((\infty, -1)\) and \((6, \infty)\).

Important details of interval notation include:

• Parentheses \(()\): Used to show that endpoints are not included in the interval. This happens when the inequality is strict (like \(>\) or \(<\)).
• Brackets \([]\): Would be used if the endpoints were included, which happens with non-strict inequalities (like \(\geq\) or \(\leq\)), but not in this exercise.

Therefore, the solution in interval notation is \((\infty, -1) \cup (6, \infty)\). This uses a union symbol \(\cup\) to combine the intervals where the inequality is satisfied.

Graphing Inequalities on a Number Line

To visualize the solution of a quadratic inequality, sketching on a number line is very helpful. This visual representation not only confirms the solution intervals but clarifies which parts belong to the solution set.

For the inequality \(x^2 - 5x - 6 > 0\), first, mark the critical points on the number line: \(x = -1\) and \(x = 6\). Since the inequality is strict, you mark these points with open circles to indicate they are not included in the solution.

Next, shade the number line in the intervals where the inequality is true. For this exercise, shade the parts from \(-\infty\) to \(-1\) and from \(6\) to \(\infty\). This shading represents that all these \(x\) values satisfy the inequality. Using a graph can greatly improve understanding by bridging algebraic solutions with geometric representation.