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Chapter 11: Problem 103

Use inspection to describe inequality's solution set. Do not solve any of the inequalities. \(\frac{1}{(x-2)^{2}}>0\)

Short Answer

Expert verified
The solution set for this inequality is \(x\in (-\infty,2) \cup (2, +\infty)\). This means x can be any real number except 2.

Step by step solution

01

- Identify the zero of the function

The zero of this function comes from the denominator being equal to zero due to the division. If \(x-2\) is equal to 0, then x is 2 as per the null-factor rule.
02

- Check the signs of the intervals

A quadratic function is always positive when squared, except where it is undefined, which we found out in the previous step to be \(x=2\). So, for \(x<2\) and \(x>2\), the function will be positive.
03

- Define the solution set

Putting it all together, since the problem looks for a greater-than-zero solution, the solution set will basically contain all real numbers except for 2. Thus, the answer is \(x\in (-\infty,2) \cup (2, +\infty)\). However, note that this is an open interval, as the function is undefined when x equals 2.

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Key Concepts

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

Quadratic Functions
Quadratic functions are a crucial concept in algebra, often taking the form \( ax^2 + bx + c = 0 \). Quadratics are characterized by the highest degree being 2, which typically results in a parabolic graph. The defining feature of a parabola is its symmetrical shape, opening either upwards or downwards.

When exploring quadratic functions, it's important to understand their behavior:
  • If the quadratic is positive, meaning the leading coefficient \( a \) is greater than 0, the parabola opens upwards.
  • If \( a \) is negative, it opens downwards.
  • The vertex, or turning point, of the parabola highlights the maximum or minimum of the function.
  • The quadratic will be undefined at certain points, such as when attempting to evaluate expressions like \( \frac{1}{(x-2)^2} \), which is not defined when \( x = 2 \), since it leads to division by zero.
Recognizing these essential characteristics of quadratic functions can help when analyzing and solving related algebraic expressions and inequalities.
Null-Factor Rule
The null-factor rule is an invaluable tool when solving equations involving quadratic functions. This principle tells us that if a product of two factors equals zero, then at least one of the factors must also be zero. This is particularly useful in identifying the roots or zeros of quadratic equations.

To use the null-factor rule effectively, remember the following steps:
  • Express the quadratic as a multiplication of its factors, such as \( (x - p)(x - q) = 0 \).
  • Apply the rule: either \( (x - p) = 0 \) or \( (x - q) = 0 \) must be true.
  • Solve for \( x \) to find the roots separately, resulting in values \( x = p \) and \( x = q \).
This rule is a quick method for solving quadratics and other polynomial expressions, making it foundational to algebraic problem-solving. Remember that the determination of solutions using this rule assumes the entire expression equals zero initially.
Interval Notation
Interval notation is a concise way to express the range of values in a solution set, especially for inequalities. Understanding how to use interval notation can simplify the presentation of solutions, making it easier to interpret and communicate.

Here's a quick guide on how interval notation works:
  • Parentheses \(( )\) indicate that a value is not included in the interval, known as an "open interval." For example, \( (a, b) \) includes all numbers greater than \( a \) and less than \( b \), but not \( a \) or \( b \) themselves.
  • Square brackets \([ ]\) indicate an "inclusive interval," where the endpoint is included. For example, \( [a, b] \) means all numbers from \( a \) to \( b \). The values \( a \) and \( b \) are both included.
  • Combining intervals can further describe complex solution sets. For instance, \( (-\infty, a) \cup (b, \infty) \) represents all real numbers except \( a \), where the union \( \cup \) indicates the combined intervals.
Using interval notation effectively allows for clear communication of solution sets, especially within the context of inequalities and functions where certain values or ranges are excluded.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I use the square root property to determine the length of a right triangle's side, I don't even bother to list the negative square root.

A company manufactures wheelchairs. The average cost function, \(\bar{C},\) of producing \(x\) wheelchairs per month is given by $$\bar{C}(x)=\frac{500,000+400 x}{x}$$ The graph of the rational function is shown. Use the function to solve. Describe the company's production level so that the average cost of producing each wheelchair does not exceed \(\$ 410 .\) Use a rational inequality to solve the problem. Then explain how your solution is shown on the graph.

A pool has an inlet pipe to fill it and an outlet pipe to empty it. It takes 2 hours longer to empty the pool than it does to fill it. The inlet pipe is turned on to fill the pool, but the outlet pipe is accidentally left open. Despite this, the pool fills in 8 hours. How long does it take the outlet pipe to empty the pool? Round to the nearest tenth of an hour.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I need to find a square root to determine the discriminant.

Determine whether statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \((x+3)^{2}, x \neq-3,\) resulting in the equivalent inequality \((x-2)(x+3)<2(x+3)^{2}\).
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