Problem 66 - Algebra A Combined Function

Question

Solve each inequality, then use a graphing calculator to check. $$ x^{2}-4 x-5<0 $$

Step-by-Step Solution

Answer

The solution is the interval \((-1, 5)\).

Step 1: Factor the Quadratic

We start by factoring the quadratic expression. The expression \( x^2 - 4x - 5 \) can be factored into \((x - 5)(x + 1)\). This comes from finding two numbers that multiply to -5 and add to -4, which are -5 and +1.

Step 2: Find the Critical Points

Set each factor equal to zero to find the critical points of the inequality. Solve \(x - 5 = 0\) to get \(x = 5\) and \(x + 1 = 0\) to get \(x = -1\). These are the points where the expression changes sign.

Step 3: Test the Intervals

The critical points divide the number line into three intervals: \((-\infty, -1)\), \((-1, 5)\), and \((5, \infty)\). Test a number from each interval in the inequality. For \((-\infty, -1)\), test \(x = -2\), for \((-1, 5)\), test \(x = 0\), and for \((5, \infty)\), test \(x = 6\).

Step 4: Determine the Sign of the Product

When \(x = -2\), \((x - 5)(x + 1) = (-2 - 5)(-2 + 1) > 0\). When \(x = 0\), \((x - 5)(x + 1) = (0 - 5)(0 + 1) < 0\). When \(x = 6\), \((x - 5)(x + 1) = (6 - 5)(6 + 1) > 0\). Thus, the inequality \((x - 5)(x + 1) < 0\) is satisfied in the interval \((-1, 5)\).

Step 5: Graph the Solution

On a graphing calculator, plot the function \(y = x^2 - 4x - 5\). Observe where the graph is below the x-axis. The graph is below the x-axis between \(x = -1\) and \(x = 5\), confirming our solution.

Key Concepts

Factoring Quadratics

To solve quadratic inequalities, factoring quadratics is an essential first step. Let's consider the quadratic expression from our problem: \[ x^2 - 4x - 5 \]Our goal is to express it in a factored form, like \((x - a)(x - b)\), which makes identifying roots easier. Factoring involves finding two numbers that multiply to the constant term \(-5\) and add up to the middle coefficient \(-4\). These numbers are \(-5\) and \(+1\). Thus, you can write:\[ x^2 - 4x - 5 = (x - 5)(x + 1) \]Factoring transforms our inequality into:\[ (x - 5)(x + 1) < 0 \]This factored form gives us critical points, which we will discuss next.

Critical Points

In solving quadratic inequalities, finding the critical points is crucial as they determine the regions of interest on the number line. After factoring the quadratic expression, set each factor equal to zero to find these critical points:\[\]- Solve \( x - 5 = 0 \) to find \( x = 5 \). This is one critical point.- Solve \( x + 1 = 0 \) to find \( x = -1 \). This is another critical point.These points \( x = 5 \) and \( x = -1 \) are where the expression might change its sign. They divide the number line into different intervals, and we need to test these intervals to find where the inequality holds true.

Number Line Intervals

The critical points \(x = -1\) and \(x = 5\) create intervals on the number line that need testing. These intervals are:1. \((-\infty, -1)\)2. \((-1, 5)\)3. \((5, \infty)\)In each interval, choose a number and substitute it back into the inequality \((x - 5)(x + 1) < 0\) to check:- For \((-\infty, -1)\), let \(x = -2\). Calculate \((x - 5)(x + 1) > 0\), which does not satisfy the inequality.- For \((-1, 5)\), let \(x = 0\). Calculate \((x - 5)(x + 1) < 0\), which satisfies the inequality.- For \((5, \infty)\), let \(x = 6\). Calculate \((x - 5)(x + 1) > 0\), which does not satisfy the inequality.Thus, the valid solution for \((x - 5)(x + 1) < 0\) is \((-1, 5)\).

Graphing Calculator

A graphing calculator can visually confirm the solution to a quadratic inequality. For our problem, plot the function:\[ y = x^2 - 4x - 5 \]On the graphing calculator, observe where the graph is below the x-axis. The x-axis crossings are the points \(x = -1\) and \(x = 5\), exactly our critical points.- Between these points, the graph dips below the x-axis.- This visual check confirms our analytical solution that the function is less than zero from \(-1\) to \(5\).Using a graphing calculator not only verifies the solution but also provides a more intuitive understanding of the inequality's behavior across different intervals.