Problem 18 Solve the inequality. Express th... [FREE SOLUTION]

Chapter 1: Problem 18

Solve the inequality. Express the solution as an interval or as the union of intervals. Mark the solution on a number line. $$\frac{x^{2}-4 x+4}{x^{2}-2 x-3} \leq 0$$

Short Answer

Expert verified

The solution to the equation $\frac{(x-2)^{2}}{(x-3)(x+1)} \leq 0$ is $(-\infty, -1) \cup (2,3)$.

Step by step solution

Factoring the Polynomial in the Numerator and Denominator

We factor the numerator $x^{2}-4x+4$ and the denominator $x^{2}-2x-3$ of the rational expression, which gives $\frac{(x-2)^{2}}{(x-3)(x+1)}$. This rewriting makes it easier to analyze and find the zeros and undefined points of the inequality.

Find the Zeros and Undefined Points

We set the numerator equal to 0 and solve to find the zeros of the expression, which gives $x=2$. And we set the denominator equal to 0 and solve to find the undefined points of the expression, which gives $x=-1$ and $x=3$.

Create Test Points from the Relevant Intervals

The original inequality is not defined and also changes sign at $x=-1$, $x=2$, and $x=3$. We thus create four intervals, each separated by those values. Choose a test point from each interval such as -2, 0, 2.5, and 4.

Test the Intervals

Plug each test point into the simplified inequality $\frac{(x-2)^{2}}{(x-3)(x+1)} \leq 0$. This yields negative, positive, negative, and positive values respectively. So intervals where $x<-1$ and $1<x<3$ satisfy the inequality.

Final Solution

Combine the intervals and include the value where the expression equals to 0. The solution is thus $(-\infty, -1) \cup (2,3)$. To express this on a number line, draw a line marking the numbers -1, 2, and 3. Make an open circle at -1 and 3, and a closed circle at 2. Shade the area between -1 and 2, and also the area between 2 and 3.

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Key Concepts

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

Factoring Polynomials

Factoring polynomials is a crucial step in solving inequalities, particularly when dealing with rational expressions. It involves breaking down a polynomial into a product of simpler polynomials. This process can make complex expressions much easier to handle. Let's take the expression from the exercise:

The numerator is the quadratic polynomial $ x^{2} - 4x + 4 $.
Notice that this quadratic can be expressed as a perfect square: $ (x-2)^{2} $.

Similarly, the denominator $ x^{2} - 2x - 3 $ can be factored into two linear factors:

$ (x-3)(x+1) $.

Factoring helps identify the zeros and undefined points of rational expressions, helping you visually analyze the problem on a number line. A key to solving polynomial inequalities is understanding that these factored forms reveal intervals where the expression changes sign.

Rational Expressions

When solving inequalities involving rational expressions, it is important to comprehend how they function. A rational expression is a fraction where both the numerator and the denominator are polynomials.

The given inequality involves the rational expression $ \frac{(x-2)^{2}}{(x-3)(x+1)} \leq 0 $.
Critical aspects are the zeros of the numerator (solutions making the expression zero) and undefined values from the denominator (solutions that make the denominator zero).
This particular expression is zero at $ x=2 $, and is undefined at $ x=-1 $ and $ x=3 $.

Analyzing rational expressions includes recognizing where they change signs, especially where they are undefined. It's critical to test regions between these critical values to decide where the inequality holds true, guiding you to the solution.

Interval Notation

Interval notation is a shorthand method used to express solutions of inequalities or unions of intervals. It conveys a range of numbers and is important for simplifying complex sets that represent solutions.

For instance, the solution from the exercise is written as $(-\infty, -1) \cup (2, 3)$.
This means that $x$ is in the solution set when it is less than $-1$ or between $2$ and $3$.
The use of parentheses denotes that these endpoints are not included in the solution set. However, a closed bracket, like $[\,2]$, would mean inclusion.

When you plot this on a number line, open circles indicate non-inclusivity at points like $-1$ and $3$, while closed circles or shaded areas indicate inclusive points or ranges. Interval notation serves as an efficient and standardized way to record the solution, making it easier to understand and communicate the range or union of solutions.

91影视

Solve the inequality. Express the solution as an interval or as the union of intervals. Mark the solution on a number line. $$\frac{x^{2}-4 x+4}{x^{2}-2 x-3} \leq 0$$

Short Answer

Step by step solution

Factoring the Polynomial in the Numerator and Denominator

Find the Zeros and Undefined Points

Create Test Points from the Relevant Intervals

Test the Intervals

Final Solution

Key Concepts

Factoring Polynomials

Rational Expressions

Interval Notation

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