Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$x^{2}-6 x+9<16$$

Step-by-Step Solution

Answer

The solution to the inequality \(x^{2}-6 x+9<16\) is \(x<7\) and \(x>-1\), which can be verified graphically

Step 1: Rearrange the inequality

We first need to rearrange the inequality to set it to less than zero, by subtracting 16 from both sides. The inequality then becomes \(x^{2}-6x+9-16<0\), which simplifies to \(x^{2}-6x-7<0\)

Step 2: Factor the quadratic expression

Factoring the expression \(x^{2}-6x-7\) we get \((x-7)(x+1)<0\). The factors are \(x-7\) and \(x+1\).

Step 3: Find the roots of the quadratic expression

Set each factor equal to zero and solve for \(x\) to find the roots of the quadratic expression. The roots are \(x=7\) and \(x=-1\).

Step 4: Test intervals between the roots

We then test the sign of the expression in the intervals between roots. Any number less than -1, between -1 and 7, and any number greater than 7 will be tested. Choose test points such as -2, 0, and 8. If the inequality is satisfied, the interval is part of the solution set.

Step 5: Draw the solution on the real number line

Mark the roots -1 and 7 on the number line. Then, according to the intervals that satisfy the inequality, draw a line or a point above the number.

Step 6: Verify the solution graphically

Use a graphing utility to plot the quadratic equation \(y = x^{2}-6x-7\). The solution of the inequality corresponds to where the curve is below the x-axis. This should match the solution obtained above.

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is essential for solving inequalities because it helps us determine the roots or solutions of the equation. The given quadratic inequality is \(x^2 - 6x - 7 < 0\). To factor this, we look for two numbers whose product is \(-7\) and whose sum is \(-6\). These numbers are \(-7\) and \(1\). Thus, the factored form is \((x - 7)(x + 1)\).

• Factoring helps break down complex expressions into simpler products.
• Finding the roots becomes straightforward once the expression is factored.

Once factored, you can quickly determine where the expression equals zero, which in this case, is at \(x = 7\) and \(x = -1\). These are the roots of the quadratic expression. Understanding these roots is crucial for analyzing the solution set of the inequality.

Graphing Inequalities

Graphing inequalities involves illustrating which parts of the number line or plane satisfy the inequality. In this context, after factoring the quadratic expression, you have \((x - 7)(x + 1) < 0\). This expression represents areas where the product of the terms is negative.

• Plot the roots on the graph; here at \(x = 7\) and \(x = -1\).
• Test the intervals between these roots to find where the inequality holds true.
• Choose sample points in each interval, such as \(-2\), \(0\), and \(8\).

By inserting these test points into the inequality, you can determine the kind of interval that satisfies the inequality. For example, testing the point \(0\) in \((x - 7)(x + 1) < 0\) gives a negative product, indicating this part of the graph satisfies the inequality.

Real Number Line

The real number line is a visual tool used for graphing the solutions to inequalities. When solving the quadratic inequality \((x - 7)(x + 1) < 0\), you use the real number line to mark the points \(x = 7\) and \(x = -1\).

• Mark the roots clearly on the line to set boundaries.
• Identify the intervals on the line between and beyond the roots.
• Shade or highlight the sections where the inequality is satisfied.

After testing the intervals, you'll notice that the interval between \(x = -1\) and \(x = 7\) needs shading, as this matches where the original inequality \(x^2 - 6x - 7 < 0\) holds true. These shaded intervals represent all the \(x\) values that satisfy the inequality.

Verifying Solutions Graphically

Verifying solutions graphically involves using graphing tools or utilities to ensure the accuracy of your solution to a quadratic inequality. For the inequality \(x^2 - 6x - 7 < 0\), graph the expression \(y = x^2 - 6x - 7\) and observe the behavior of the curve.

• Observe where the curve dips below the x-axis to identify where the inequality is satisfied.
• Ensure it matches the previously shaded portion on the real number line from graphic or test intervals.
• Graphical visualization provides a more intuitive understanding of solutions.

When graphing \(y = x^2 - 6x - 7\), you'll see that the curve dips below the x-axis between \(x = -1\) and \(x = 7\). This confirms that the inequality is satisfied in this interval on the real number line, validating your solution.