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Chapter 7: Problem 50

Add or subtract as indicated. Simplify the result, if possible. $$\frac{3 x+7}{x^{2}-5 x+6}-\frac{3}{x-3}$$

Short Answer

Expert verified
\(\frac{13}{(x-3)(x-2)}\)

Step by step solution

01

Factor the Denominator

Factor the denominator of the first fraction \(x^{2}-5 x+6\) to \((x-3)(x-2)\). So, the fraction can be written as \(\frac{3 x+7}{(x-3)(x-2)}\).
02

Identify Common Denominator

The denominators of the two expressions are \((x-3)(x-2)\) and \(x-3\). The least common denominator (LCD) is \( (x-3)(x-2)\). So rewrite the subtracted fraction with this denominator as \(\frac{3(x-2)}{(x-3)(x-2)}\). The expression then becomes \(\frac{3 x+7}{(x-3)(x-2)}-\frac{3(x-2)}{(x-3)(x-2)}\).
03

Subtract Numerators

Since the fractions have the same denominator, subtract the numerators: \(\frac{3 x+7-3(x-2)}{(x-3)(x-2)} = \frac{3x+7-3x+6}{(x-3)(x-2)} = \frac{13}{(x-3)(x-2)}\) (You can verify that the numerator can't be simplified further by performing the polynomial subtraction).
04

Final Answer

Thus, the expression is simplified to \(\frac{13}{(x-3)(x-2)}\).

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.. $$Find the missing polynomials: $\quad-\frac{3 x-12}{2 x}=\frac{3}{2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x-5}{6} \cdot \frac{3}{5-x}=\frac{1}{2}\( for any value of \)x$ except 5$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x-y}+\frac{y}{y-x}$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{6}{x-1}-\frac{5}{1-x}$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{x}{x+6}-1$$
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