Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(x^{2}<4 x+21\)

Short answer

The solution is \( x \in (-3, 7) \).

Step-by-step solution

Rearrange the Inequality

Start by rearranging the given inequality to set it to zero on one side. Subtract 4x and 21 from both sides:\( x^2 - 4x - 21 < 0 \)

Factor the Quadratic Expression

Factor the quadratic expression on the left-hand side. We look for two numbers that multiply to -21 and add to -4:\( x^2 - 4x - 21 = (x - 7)(x + 3) \)

Determine the Critical Points

Set each factor equal to zero and solve for x to find the critical points:\( x - 7 = 0 \) gives \( x = 7 \)\( x + 3 = 0 \) gives \( x = -3 \)

Test Intervals

Use the critical points to divide the number line into intervals: \((-\infty, -3)\), \((-3, 7)\), and \((7, \infty)\). Test a point from each interval in the inequality \( (x - 7)(x + 3) < 0 \) to see if the inequality is satisfied.

Determine the Sign of Each Interval

For \( x = -4 \) in \( (-\infty, -3) \):\( (-4 - 7)(-4 + 3) = (-11)(-1) = 11 > 0 \), not valid.For \( x = 0 \) in \( (-3, 7) \):\( (0 - 7)(0 + 3) = (-7)(3) = -21 < 0 \), valid.For \( x = 8 \) in \( (7, \infty) \):\( (8 - 7)(8 + 3) = (1)(11) = 11 > 0 \), not valid.

Combine the Valid Intervals

Combine the intervals where the inequality is satisfied. Since only the interval \( (-3, 7) \) is valid, this is the solution to the inequality.

Key concepts

factoring quadratic expressions

Factoring quadratic expressions is a key step when dealing with quadratic inequalities. It lets us split the quadratic into simpler factors. For instance, with the inequality \(x^2 - 4x - 21 < 0\), we turn the quadratic into two factors: \((x - 7)(x + 3)\). This simplifies further solving.

Factoring involves finding numbers that multiply to give the constant term (in this case, -21) while adding to the coefficient of the linear term (here, -4). It's like a puzzle. For \(x^2 - 4x - 21\), these numbers are -7 and 3, giving us \((x - 7)(x + 3)\).

To factor, always look for pairs of numbers that satisfy both conditions. If factoring seems tough, remember to try different pairs until you find the right one. Practice helps you get better at spotting the factors quickly.

interval testing

Interval testing checks which parts of a number line make the inequality true. After factoring, we get: \((x - 7)(x + 3) < 0\). Next, we find critical points (where the factors are zero). These split the number line into intervals.

Here, the critical points are \(x = -3\) and \(x = 7\). So our intervals are: \((-\text{infinity}, -3)\), \((-3, 7)\), and \((7, \text{infinity})\).

To test intervals, pick a number inside each interval and plug it into the inequality to see if it’s true. For \((-3, 7)\), using \(x = 0\):

• \((0 - 7)(0 + 3) = -21\), which is less than 0, so it’s valid.

Repeat for each interval to see where the inequality holds. Interval testing helps confirm which ranges fit the solution.

critical points

Critical points are essential in solving quadratic inequalities. They divide the number line into intervals that can be tested. To find critical points, set each factor to zero and solve for \(x\).

For example: From \((x - 7)(x + 3) < 0\), we get:

• \(x - 7 = 0\), which gives \(x = 7\)
• \(x + 3 = 0\), which gives \(x = -3\)

These points \(x = 7\) and \(x = -3\) are our critical points. They help in forming intervals for further testing. Critical points mark where the expression changes behavior (from positive to negative or vice versa). Always solve for them accurately as they impact the intervals and final solution.

number line analysis

Number line analysis is used to visualize and solve inequalities easily. After finding and testing intervals, we sketch a number line to show our results.

Start by marking the critical points on the number line. In our example, place points at \(x = -3\) and \(x = 7\). Next, use interval testing results:

• Interval \((-\infty, -3)\): Not valid
• Interval \((-3, 7)\): Valid
• Interval \((7, \text{infty})\): Not valid

Highlight the valid interval \((-3, 7)\) on the number line. Number line analysis gives a quick visual answer and makes it easier to understand and interpret the solution set. This technique can be applied to different inequalities and can simplify understanding and solving them.