Problem 82
Question
In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{3}{x-4}=\frac{?}{4-x}$$
Step-by-Step Solution
Verified Answer
? = -3
1Step 1: Recognize equivalent expressions
Identify the expressions in the denominators: \(x-4\) and \(4-x\). Notice that \(4-x = -(x-4)\).
2Step 2: Simplify using the property
Since \(4-x = -(x-4)\), rewrite the right-hand side of the equation as: \( \frac{?}{-(x-4)} \).
3Step 3: Multiply numerator and denominator by -1
To make the denominators exactly equal, consider multiplying the numerator and the denominator of the right-hand side of the equation by -1: \( \frac{?}{-(x-4)} = \frac{-?}{x-4} \).
4Step 4: Equate the numerators
Since the denominators are now the same, the numerators must be equal too: \(3 = -?\). Therefore, \? = -3\.
Key Concepts
Identifying Equivalent ExpressionsSimplifying Rational ExpressionsProperties of DenominatorsMultiplying Numerators and Denominators
Identifying Equivalent Expressions
When working with rational expressions, a critical skill is recognizing when two expressions are equivalent. This means they represent the same value for all values of the variable, where they are defined.
In the given exercise, we’re dealing with the expressions in the denominators: \((x-4)\) and \((4-x)\). Recognizing that these expressions can be transformed into one another is a crucial step.
Notice that:
In the given exercise, we’re dealing with the expressions in the denominators: \((x-4)\) and \((4-x)\). Recognizing that these expressions can be transformed into one another is a crucial step.
Notice that:
- \((4-x)\) is essentially reversed with a subtraction in the opposite order from \((x-4)\).
- We can express this as: \((4-x) = -(x-4)\). This transformation shows that they are equivalent, except for a sign change.
Simplifying Rational Expressions
Simplifying rational expressions helps in making them easier to work with. It involves reducing the expression to its simplest form where the numerator and denominator are as simple as possible.
Next, to simplify further, we note that multiplying the numerator and denominator by -1 doesn’t change the value of the expression:
- In the example, we start with \(\frac{3}{x-4} = \frac{?}{4-x}\).
- Recognizing that \((4-x) = -(x-4)\), we can rewrite the right-hand side:
- \( \frac{?}{-(x-4)} \).
Next, to simplify further, we note that multiplying the numerator and denominator by -1 doesn’t change the value of the expression:
- \( \frac{?}{-(x-4)} = \frac{-?}{x-4} \).
- By doing so, the denominators align perfectly, and the equivalence of the expressions becomes apparent.
Properties of Denominators
Denominators have unique properties that can be used for simplification. Understanding these properties can ease the process of working with rational expressions.
This step is pivotal as it illuminates the power of properties of denominators in manipulating rational expressions to achieve equivalence or to simplify them.
- A denominator dictates where the rational expression is undefined, as the expression cannot have zero as a denominator.
- The product or ratio involving negative terms can transform denominators and affect the overall expression without changing its value.
- In our problem, acknowledging \((4-x) = -(x-4)\) allows us to rewrite the denominator in a useful form:
- \(\frac{?}{-(x-4)} \).
This step is pivotal as it illuminates the power of properties of denominators in manipulating rational expressions to achieve equivalence or to simplify them.
Multiplying Numerators and Denominators
Multiplying the numerator and denominator by the same value is an essential technique in algebra.
This process, called multiplying by the multiplicative identity (which is 1), lets us adjust parts of the expression without changing its overall value.
In our exercise, we need to align the denominators to identify equivalent expressions:
Here, multiplying by -1 effectively changes the sign, aligning the denominator properly:
This process, called multiplying by the multiplicative identity (which is 1), lets us adjust parts of the expression without changing its overall value.
In our exercise, we need to align the denominators to identify equivalent expressions:
- Recognize the expression: \(\frac{?}{-(x-4)} \).
- Multiply the numerator and denominator by -1:
- \(\frac{?}{-(x-4)} = \frac{-?}{x-4} \).
Here, multiplying by -1 effectively changes the sign, aligning the denominator properly:
- Since the denominators are now the same, we can equate the numerators.
- \(3 = -?\) leads to \(? = -3\).
Other exercises in this chapter
Problem 82
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