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Chapter 1: Problem 3

Determine whether the given value is a solution of the equation. \(\frac{1}{x}-\frac{1}{x-4}=1\) (a) \(x=2\) (b) \(x=4\)

Short Answer

Expert verified
(a) x=2 is a solution; (b) x=4 is not a solution.

Step by step solution

01

Substitute and Simplify the Equation for x=2

Substitute \(x=2\) into the equation \(\frac{1}{x}-\frac{1}{x-4}=1\). This gives us \(\frac{1}{2}-\frac{1}{-2}=1\). Simplifying the right side results in \(\frac{1}{2}+\frac{1}{2}=1\), which simplifies to \(1=1\). This is true, so \(x=2\) is a solution.
02

Substitute and Check x=4 for Undefined Operation

Substitute \(x=4\) into the equation \(\frac{1}{x}-\frac{1}{x-4}=1\). This gives \(\frac{1}{4}-\frac{1}{0}=1\). Since division by zero is undefined, \(x=4\) cannot be a solution.

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Key Concepts

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

Understanding the Substitution Method
The substitution method is a way to find whether or not a specific value is a solution to an equation. This technique involves replacing a variable in the equation with a given number to see if it makes the equation true. Let's break it down using the example provided.When you have an equation like \( \frac{1}{x} - \frac{1}{x-4} = 1 \), and you're given a value such as \( x = 2 \), you substitute 2 in place of \( x \) in the equation. You'll get \( \frac{1}{2} - \frac{1}{2-4} = 1 \), which simplifies to \( \frac{1}{2} + \frac{1}{2} = 1 \).
  • This simplification checks the equality of the left-hand side to the right-hand side of the equation.
  • If both sides are equal, as in this case \(1=1\), the substitution confirms that \( x=2 \) is a solution.
When using substitution, always ensure you substitute correctly and simplify the expression.
Step-by-step Solution Verification
After substituting a value in the equation and simplifying, the next important step is solution verification. This process ensures the solution satisfies the original equation without any errors in calculation. Verification involves:
  • Double-checking your substitution by ensuring the operation per each term is precise.
  • Confirming that your arithmetic calculations are correct.
For example, after substituting \(x=2\) in \(\frac{1}{x} - \frac{1}{x-4} = 1\), verifying involves ensuring that \(\frac{1}{2} + \frac{1}{-2} = 0\) was correctly simplified to \(1 = 1\). This confirms that our solution was correct. Verification prevents errors that can occur if steps are miscalculated or misunderstood.
Recognizing Undefined Operations
In mathematics, certain operations are not allowed; division by zero is one of them. Recognizing undefined operations is crucial when solving equations.When substituting \(x = 4\) into the equation \(\frac{1}{x} - \frac{1}{x-4} = 1\), we end up with \(\frac{1}{4} - \frac{1}{0}\).
  • Since division by zero is undefined, the equation becomes impossible to solve for \(x = 4\).
  • Mathematically, division by zero does not produce a valid number, causing any equation containing it to be invalid.
Understanding this concept helps you avoid potential pitfalls and correctly identify a value that cannot be a solution. Always check for undefined operations when substituting values.

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Most popular questions from this chapter

(a) Show that \(a b=\frac{1}{2}\left[(a+b)^{2}-\left(a^{2}+b^{2}\right)\right]\) (b) Show that \(\left(a^{2}+b^{2}\right)^{2}-\left(a^{2}-b^{2}\right)^{2}=4 a^{2} b^{2}\) (c) Show that \(\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=(a c+b d)^{2}+(a d-b c)^{2}\) (d) Factor completely: \(4 a^{2} c^{2}-\left(a^{2}-b^{2}+c^{2}\right)^{2}\)

Use a Special Factoring Formula to factor the expression. $$(x+3)^{2}-4$$

Sketch the region in the coordinate plane that satisfies both the inequalities \(x^{2}+y^{2} \leq 9\) and \(y \geq|x| .\) What is the area of this region?

Use a Special Factoring Formula to factor the expression. $$27 x^{3}+y^{3}$$

Make up several pairs of polynomials, then calculate the sum and product of each pair. Based on your experiments and observations, answer the following questions. (a) How is the degree of the product related to the degrees of the original polynomials? (b) How is the degree of the sum related to the degrees of the original polynomials?
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