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Magazine Chapter 11: Problem 77
Solve. $$x^{2}-5 x-14<0$$
Short Answer
Expert verified
The solution to the inequality is \(-2 < x < 7\).
Step by step solution
01
Set the inequality to an equation
To solve the inequality, first consider the related equation: \[x^2 - 5x - 14 = 0\].
02
Factor the quadratic equation
Next, factor the quadratic equation. We need to find two numbers that multiply to -14 and add up to -5. These numbers are 7 and -2. So, we can write:\[x^2 - 5x - 14 = (x - 7)(x + 2)\].
03
Find the roots
Set each factor equal to zero to find the roots:\(x - 7 = 0\) and \(x + 2 = 0\).Solving these, we get:\[x = 7\] and \[x = -2\].
04
Determine the test intervals
The roots divide the number line into three intervals. These intervals are: \((-\infty, -2)\), \((-2, 7)\), and \((7, \infty)\).
05
Test intervals in the original inequality
Pick a test point from each interval and substitute it into the inequality \(x^2 - 5x - 14 < 0\):1. For the interval \((-\infty, -2)\), pick \(x = -3\): \((-3)^2 - 5(-3) - 14 = 9 + 15 - 14 = 10 > 0\).2. For the interval \((-2, 7)\), pick \(x = 0\): \(0^2 - 5(0) - 14 = -14 < 0\).3. For the interval \((7, \infty)\), pick \(x = 8\): \(8^2 - 5(8) - 14 = 64 - 40 - 14 = 10 > 0\).
06
Combine test results
From the test results, the original inequality \(x^2 - 5x - 14 < 0\) holds for \(-2 < x < 7\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a 鈮� 0\). These equations are called 'quadratic' because of the \(x^2\) term. Solving quadratic equations can be done through various methods such as factoring, using the quadratic formula, or completing the square.
- The quadratic equation related to our exercise is \(x^2 - 5x - 14 = 0\).
- Our goal is to find the values of \(x\) that satisfy this equation, which helps us in solving the inequality.
Factoring
Factoring involves expressing a quadratic equation as a product of its linear factors. For the equation \(x^2 - 5x - 14 = 0\), we look for two numbers that multiply to \(-14\) and add to \(-5\). These numbers are \(7\) and \(-2\).
By rewriting the quadratic expression, we get:
\((x - 7)(x + 2)\).
By rewriting the quadratic expression, we get:
\((x - 7)(x + 2)\).
- This means that \(x = 7\) or \(x = -2\) are the solutions to the equation.
- Factoring is useful because it simplifies quadratic equations into a product of simpler expressions.
Number Line Intervals
Once we have the solutions \(x = 7\) and \(x = -2\), these values divide the number line into intervals. These intervals are useful for testing where the original inequality holds true.
We consider the following intervals:
We then pick a test point from each interval and substitute it back into our original inequality to check if the inequality is satisfied.
We consider the following intervals:
- \((-\finity, -2)\)
- \((-2, 7)\)
- \((7, \infty)\)
We then pick a test point from each interval and substitute it back into our original inequality to check if the inequality is satisfied.
Inequalities
To solve a quadratic inequality like \(x^2 - 5x - 14 < 0\), we analyze different intervals on the number line created by the roots of the related quadratic equation. Here's how:
From the test results, we see the inequality \(x^2 - 5x - 14 < 0\) is true in the interval \((-2, 7)\). Thus, the solution to the inequality is \(-2 < x < 7\).
- For interval \((-\finity, -2)\), pick \(x = -3\): \( (-3)^2 - 5(-3) - 14 = 10 > 0 \).
- For interval \((-2, 7)\), pick \(x = 0\): \( 0^2 - 5(0) - 14 = -14 < 0 \).
- For interval \((7, \infty)\), pick \(x = 8\): \( 8^2 - 5(8) - 14 = 10 > 0 \).
From the test results, we see the inequality \(x^2 - 5x - 14 < 0\) is true in the interval \((-2, 7)\). Thus, the solution to the inequality is \(-2 < x < 7\).
