Question

In Problems \(7-22,\) solve each inequality. 7\. \(x^{2}-3 x-10<0\)

Short answer

The solution is  \((-2, 5)\).

Step-by-step solution

- Find the Roots of the Quadratic Equation

First, solve the equation  \(x^{2}-3x-10=0\) to find its roots using the quadratic formula:  \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). In this case,  \(a=1\),  \(b=-3\), and  \(c=-10\).

- Perform the Calculations

Substitute the values into the formula:  \(x = \frac{-(-3) \pm \sqrt{(-3)^{2}-4 \cdot 1 \cdot (-10)}}{2 \cdot 1}\). This simplifies to  \(x = \frac{3 \pm \sqrt{9+40}}{2}\) and further to  \(x = \frac{3 \pm \sqrt{49}}{2}\). Thus, the roots are  \(x = \frac{3 + 7}{2} = 5\) and  \(x = \frac{3 - 7}{2} = -2\).

- Determine the Intervals to Test

The roots divide the number line into three intervals:  \((-\infty, -2)\),  \((-2, 5)\), and  \((5, \infty)\). Test a point from each interval to see where the original inequality  \(x^{2}-3x-10<0\) is satisfied.

- Test Intervals

Select a test point from each interval:- For  \((-\infty, -2)\), choose  \(x = -3\):  \((-3)^{2} - 3(-3) - 10 = 9 + 9 - 10 = 8\ (not \ < 0)\).- For  \((-2, 5)\), choose  \(x = 0\):  \(0^{2} - 3 \cdot 0 - 10 = -10\ (< 0)\).- For  \((5, \infty)\), choose  \(x=6\):  \(6^{2} - 3 \cdot 6 - 10 = 36 - 18 - 10 = 8\ (not \ < 0)\).

- Conclusion

Since the inequality  \(x^{2}-3x-10<0\) is satisfied in the interval  \((-2, 5)\), this is the solution set for the inequality.

Key concepts

quadratic formula

To solve quadratic inequalities, we first need to find the roots of the corresponding quadratic equation. The quadratic formula is an essential tool for this task. We use it when we cannot factorize the quadratic equation easily.

The quadratic formula is:
\(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\)

In the formula, \(a\) is the coefficient of \(x^2\), \(b\) the coefficient of \(x\), and \(c\) the constant term.

For instance, to solve the quadratic equation \(x^{2}-3x-10=0\), we identify
\(a=1\), \(b=-3\), and \(c=-10\).

Substituting these values into the formula, we get:

\[x = \frac{-(-3) \pm \sqrt{(-3)^{2}-4 \cdot 1 \cdot (-10)}}{2 \cdot 1}\]

Simplifying this gives us the roots designed to isolate the critical points of the inequality.

inequality intervals

After finding the roots of the quadratic equation, we can use these roots to divide the number line into intervals. This helps determine where the inequality holds true.

For the inequality \(x^{2}-3x-10<0\), the roots are \(x=5\) and \(x=-2\). These roots split the number line into three distinct intervals:

• \((-\thspace{-0.25em}\ttinfty, -2)\)
• \((-2, 5)\)
• \((5, \ttinfty)\)

The goal is to find in which of these intervals the original inequality is satisfied.

testing intervals

Testing intervals involve choosing a point within each interval to see if it satisfies the original inequality.

For the intervals resulting from our example:\(x^{2}-3x-10<0\), test points need to be selected:

• For the interval \((-\thspace{-0.25em}\ttinfty, -2)\), choose \(x=-3\). Substituting \(-3\) into the inequality gives
  \((-3)^{2}-3(-3)-10=8\). This result is not less tħan 0, so the inequality does not hold in this interval.


• For the interval \((-2, 5)\), choose \(x=0\). Substituting 0 into the inequ\-ity gives \(0^{2}-3\cdot0-10=-10\). This result is less than 0, indicating the inequality holds in this interval.
  
  
• For the interval \((5, \ttinfty)\), choose \(x=6\), substituting into the inequality gives
  \(6^{2}-3\cdot6-10=8\). This result is no -t less than 0, showing the inequality does not hold in this interval.

By testing points from each interval, you determine that \((-2,5)\)
is where the severe inequality \(x^{2}-3x-10<0\) holds true.

roots of quadratic equations

In solving quadratic inequalities, finding the roots of the equivalent quadratic equation is critical. These roots, also called zeroes or solutions, give points where the graph of the quadratic function intercepts the x-axis. These interception points help segment the number line and identify the intervals where the inequality may hold.

For the example \(x^{2}-3x-10=0\), using the quadratic formula gives
the roots \(x=5\) and \(x=-2\).

These roots mean that the quadratic equation's graph intercepts
the x-axis at points 5 and -2. Representing visually, we can then
test the intervals divided by these root points to find the intervals
where our original inequality holds true. In this exercise,
the inequality \(x^{2}-3x-10<0\) holds between the intervaly< \((-2,5)\)