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Chapter 2: Problem 16

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

Short Answer

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

Step by step solution

01

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 \((x - 3)(x + 3) < 0\).
02

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.
03

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.
04

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 '鈮�0', the values -3 and 3 are not included in the solution. So, the solution is \(-3 < x < 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In Exercises \(67-86,\) find expressions for \((f \circ g)(x)\) and \((g \circ f)(x)\) Give the domains of \(f \circ g\) and \(g \circ f\). $$f(x)=\frac{-x+1}{2 x+3} ; g(x)=\frac{1}{x^{2}+1}$$

Attendance at Broadway shows in New York can be modeled by the quadratic function \(p(t)=0.0489 t^{2}-0.7815 t+10.31,\) where \(t\) is the number of years since 1981 and \(p(t)\) is the attendance in millions. The model is based on data for the years \(1981-2000 .\) (Source: The League of American Theaters and Producers, Inc.) (a) Use this model to estimate the attendance in the year \(1995 .\) Compare it to the actual value of 9 million. (b) Use this model to predict the attendance for the year 2006 (c) What is the vertex of the parabola associated with the function \(p\), and what does it signify in relation to this problem? (d) Would this model be suitable for predicting the attendance at Broadway shows for the year \(2025 ?\) Why or why not? (e) \(\quad\) Use a graphing utility to graph the function \(p\) What is an appropriate range of values for \(t ?\)

Use the intersect feature of your graphing calculator to explore the real solution(s), if any, of \(x^{2}=x+k\) for \(k=0, k=-\frac{1}{4},\) and \(k=-3 .\) Also use the zero feature to explore the solution(s). Relate your observations to the quadratic formula.

In Exercises \(105-110\), find the difference quotient \(\frac{f(x+h)-f(x)}{h}\) \(h \neq 0,\) for the given function \(f\). $$f(x)=-x^{2}+x$$

The number of copies of a popular mystery writer's newest release sold at a local bookstore during each month after its release is given by \(n(x)=-5 x+100\) The price of the book during each month after its release is given by \(p(x)=-1.5 x+30 .\) Find \((n p)(3) .\) Interpret your results.
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