Problem 16 - Precalculus : Building Concepts and Connections

Question

Solve the inequality by factoring. $$x^{2}-9<0$$

Step-by-Step Solution

Answer

The solution to the inequality \(x^2 - 9 < 0\) is \(-3 < x < 3\).

Step 1: Factor Quadratic Expression

The given inequality is \(x^2 - 9 < 0\). This can be factored using the difference of squares formula, which is \(a^2 - b^2 = (a - b)(a + b)\). Therefore, \(x^2 - 9 = (x - 3)(x + 3)\). The inequality becomes \((x - 3)(x + 3) < 0\).

Step 2: Find Critical Points

Set each factor equal to zero and solve for \(x\). When \(x - 3 = 0\), \(x = 3\). When \(x + 3 = 0\), \(x = -3\). These are the critical points.

Step 3: Determine the Sign of Each Interval

Because the inequality is less than zero, we are looking for intervals where the expression is negative. Test a number in each interval: \((-\infty, -3)\), \((-3, 3)\), and \((3, \infty)\). Choose numbers within these intervals and plug it into the factored form of the quadratic expression. The expression is negative in the interval \((-3, 3)\). Specifically, when you plug in 0, a number in the interval \((-3, 3)\), the factored form, \((x - 3)(x + 3)\) becomes \(-9\), which is less than zero.

Step 4: Write the Solution

The solution to the inequality is values of \(x\) within the interval \((-3, 3)\). However, since the initial inequality is '<0' and not '≤0', the values -3 and 3 are not included in the solution. So, the solution is \(-3 < x < 3\).

Key Concepts

Quadratic Factoring

Quadratic factoring is a technique used to simplify quadratic expressions by rewriting them as products of simpler expressions. In the context of inequalities like \(x^2 - 9 < 0\), this process helps in identifying the solutions. Quadratics are often factorable by recognizing patterns such as the difference of squares. This particular format, \(a^2 - b^2\), can be rewritten as two binomial expressions: \((a - b)(a + b)\).

• Rewrite \(x^2 - 9\) as \((x - 3)(x + 3)\).
• The factored version is simpler for determining intervals where the inequality holds.

Factoring transforms the problem into a form that's easier to analyze and solve. Recognizing such patterns is a valuable skill for solving quadratic inequalities.

Difference of Squares

The concept of difference of squares is essential when dealing with quadratic expressions like \(x^2 - 9\). The difference of squares formula states that any expression like \(a^2 - b^2\) can be expressed as a product of two binomials: \((a-b)(a+b)\). This method is especially useful in simplifying expressions for inequalities or equations.

• Given \(x^2 - 9\), recognize it as \((x)^2 - (3)^2\).
• Apply the difference of squares: \((x - 3)(x + 3)\).

By applying this formula, we break down more complex expressions into manageable binomial forms. This step is crucial before proceeding to solve the inequality by finding critical points.

Critical Points

Critical points are where the factors of the quadratic expression equal zero. These points are vital in understanding where the expression changes sign, which is essential in solving inequalities. When you identify critical points, you effectively pinpoint boundaries for different intervals on the number line.

• Set each factor equaling zero: \(x - 3 = 0\) gives \(x = 3\); \(x + 3 = 0\) gives \(x = -3\).
• These points divide the number line into intervals: \((-\infty, -3)\), \((-3, 3)\), \((3, fin\infty)\).

By locating these critical points, you can easily determine where the original quadratic expression is negative, positive, or zero. They serve as markers that guide the process of interval testing.

Interval Testing

Interval testing is the process used after determining critical points to find where the inequality holds true. Each interval formed by critical points needs testing to see whether the quadratic expression is positive or negative within that segment.

• Choose a test point from each interval, such as \(-4\), \(0\), and \(4\) from \((-\infty, -3)\), \((-3, 3)\), and \((3, \infty)\) respectively.
• Substitute these points into the factored inequality \((x - 3)(x + 3) < 0\).
• For \(0\), within \((-3, 3)\), \((0 - 3)(0 + 3) = -9\), verifying the expression is negative.

This technique reveals in which intervals the inequality is valid. Ultimately, this allows us to write the solution as \(-3 < x < 3\), providing a clear resolution to the inequality.