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Chapter 2: Problem 3

Write each of the expressions as a single fraction. $$ \frac{1}{x-2}-\frac{1}{x-3} $$

Short Answer

Expert verified
Question: Combine the fractions \(\frac{1}{x-2}\) and \(\frac{-1}{x-3}\) into a single fraction. Answer: \(\frac{-1}{(x-2)(x-3)}\)

Step by step solution

01

Find the common denominator

To find the common denominator of both fractions, we can simply multiply the denominators together. In this case, the common denominator will be \((x-2)(x-3)\).
02

Rewrite the fractions with the common denominator

Now, rewrite the given expression using the common denominator: $$ \frac{1(x-3)}{(x-2)(x-3)} - \frac{1(x-2)}{(x-2)(x-3)} $$
03

Subtract the numerators of the fractions

In this step, subtract the numerators while keeping the common denominator: $$ \frac{(x-3) - (x-2)}{(x-2)(x-3)} $$
04

Simplify the numerator

Now, simplify the numerator of the fraction: $$ \frac{x - 3 - x + 2}{(x-2)(x-3)} $$ After simplifying, the expression becomes: $$ \frac{-1}{(x-2)(x-3)} $$ The final result, as a single fraction, is: $$ \frac{-1}{(x-2)(x-3)} $$

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Most popular questions from this chapter

Explain how the distributive law \(a(b+\) \(c)=a b+a c\) has been used in the identity. \(x^{2}(x+r+3)=x^{2}(x+r)+3 x^{2}\)

A contractor is managing three different job sites. It costs her $$\$ c$$ to employ a carpenter, $$\$ p$$ to employ a plumber, and $$\$ e$$ to employ an electrician. The total cost to employ carpenters, plumbers, and electricians at each site is Cost at site \(1=12 c+2 p+4 e\) Cost at site \(2=14 c+5 p+3 e\) Cost at site \(3=17 c+p+5 e\). Write expressions in terms of \(c, p,\) and \(e\) for: (a) The total employment cost for all three sites. (b) The difference between the employment cost at site 1 and site 3 . (c) The amount remaining in the contractor's budget after accounting for the employment cost at all three sites, given that originally the budget is \(50 c+10 p+20 e\)

If \(a+b+c=12\), find the value of $$(a+5)+(b-3)+(c+8)$$

If \(a-b+c=17\), what is \(2(a+1)-(b+3)+(2 c-b)\) ?

Write each expression as a sum, difference, or product of two or more algebraic fractions. There is more than one correct answer. Assume all variables are positive. $$ \frac{2 x h+h^{2}}{h} $$
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