Problem 6 - Algebra 2

Question

Solve each inequality using a graph, a table, or algebraically. $$ x^{2}-6 x-7<0 $$

Step-by-Step Solution

Answer

The solution is \(-1 < x < 7\).

Step 1: Identify the Inequality Expression

The inequality given is \( x^2 - 6x - 7 < 0 \). We need to find the values of \( x \) for which this quadratic expression is less than zero.

Step 2: Solve the Related Quadratic Equation

First, find the roots of the quadratic equation \( x^2 - 6x - 7 = 0 \). Use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -6 \), \( c = -7 \).

Step 3: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \):\[ (-6)^2 - 4 \times 1 \times (-7) = 36 + 28 = 64 \].Since the discriminant is positive, there are two real roots.

Step 4: Find the Roots

Apply the quadratic formula:\[ x = \frac{-(-6) \pm \sqrt{64}}{2 \times 1} = \frac{6 \pm 8}{2} \].This gives the roots: \( x = 7 \) and \( x = -1 \).

Step 5: Analyze the Sign of the Quadratic

The parabola described by \( x^2 - 6x - 7 \) opens upwards because the coefficient of \( x^2 \) is positive. The roots \( x = -1 \) and \( x = 7 \) divide the number line into three intervals: \((-\infty, -1)\), \((-1, 7)\), and \((7, \infty)\).

Step 6: Test the Intervals

Choose a test point from each interval and analyze the sign of the quadratic expression:- For \( x = -2 \) (in \((-\infty, -1)\)), \( (-2)^2 - 6(-2) - 7 = 4 + 12 - 7 > 0 \).- For \( x = 0 \) (in \((-1, 7)\)), \( (0)^2 - 6(0) - 7 = -7 < 0 \).- For \( x = 8 \) (in \((7, \infty)\)), \( (8)^2 - 6(8) - 7 = 64 - 48 - 7 > 0 \).

Step 7: Identify the Solution Interval

The quadratic expression is negative in the interval \((-1, 7)\). Hence, \( x^2 - 6x - 7 < 0 \) holds true for \(-1 < x < 7\).

Key Concepts

Quadratic Formula

The quadratic formula is a reliable method for solving quadratic equations. It is specifically designed to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

To apply this formula, you need to identify the values for \(a\), \(b\), and \(c\) from the equation. In the context of the inequality \(x^2 - 6x - 7 < 0\), we have:

• \(a = 1\)
• \(b = -6\)
• \(c = -7\)

Plugging these values into the quadratic formula allows us to solve for \(x\), providing the points where the quadratic graph crosses the x-axis, which are essential for determining the intervals of interest.

Discriminant

The discriminant is a key element of the quadratic formula that offers insight into the nature of the roots of a quadratic equation. It is represented as \(b^2 - 4ac\). The value of the discriminant tells us several things:

• If it is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one real root, meaning the parabola touches the x-axis at one point.
• If it is negative, there are no real roots, implying the parabola does not intersect the x-axis.

For the inequality \(x^2 - 6x - 7 < 0\), calculating the discriminant gives \(64\), a positive number. This indicates we have two real roots, which are \(x = 7\) and \(x = -1\). Understanding the discriminant is crucial as it determines how the direction of the quadratic curve behaves around these roots.

Sign Analysis

Sign analysis is a technique used to determine where a quadratic expression is positive or negative relative to its roots. This process involves examining intervals divided by the roots. For \(x^2 - 6x - 7\), the roots \(x = -1\) and \(x = 7\) divide the number line into three critical intervals:

• \((-\infty, -1)\)
• \((-1, 7)\)
• \((7, \infty)\)

By selecting a test point from each interval and evaluating the sign of the quadratic expression at these points, we can determine the sign of the expression in the entire interval:

• In \((-\infty, -1)\), the quadratic is positive.
• In \((-1, 7)\), the quadratic is negative.
• In \((7, \infty)\), the quadratic is positive again.

This analysis is essential for locating the solution intervals for inequalities.

Solution Intervals

Solution intervals represent the ranges of \(x\) values that satisfy the inequality condition. Based on our sign analysis, we know in which intervals the quadratic expression falls below zero. For the quadratic inequality \(x^2 - 6x - 7 < 0\), it is true between the roots \(-1\) and \(7\). Therefore, the solution interval is:

• \(-1 < x < 7\)

In this interval, the parabola associated with the quadratic function is located below the x-axis, fulfilling the condition of the inequality. Understanding solution intervals is critical in delineating the exact range of values that satisfy a quadratic inequality, thereby providing all possible solutions.