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Chapter 7: Problem 65

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{4 x-6}{3-2 x}$$

Short Answer

Expert verified
The simplified form of the rational expression is -2.

Step by step solution

01

Factoring out common terms

Look for any common factors in the numerator and the denominator. In this case, both \(4x - 6\) and \(3 - 2x\) have common factors. The numerator can be factored as \(2(2x - 3)\) and the denominator can be factored as \(1(3 - 2x)\)
02

Reverse the signs

The denominator has the term \(2x\) with a negative sign. To make this similar to the numerator, reverse the signs in the denominator, yielding \(-1(-3 + 2x)\). This does not change the value of the expression because multiplication by \(-1\) just changes the sign, which is equivalent to switching the order of subtraction.
03

Simplify the expression

Now the numerator and the denominator have similar terms: \(2(2x - 3)\) and \(-1(2x - 3)\). These terms can be cancelled out, so the simplified expression becomes \(\frac{-2}{1}\).

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Most popular questions from this chapter

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{3 x}{(x+1)^{2}}-\left[\frac{5 x+1}{(x+1)^{2}}-\frac{3 x+2}{(x+1)^{2}}\right]$$

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{5 x-2}{3 x-4}+\frac{2 x-3}{4-3 x}$$

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{(y-3)(y+2)}{(y+1)(y-4)}-\frac{(y+2)(y+3)}{(y+1)(4-y)}-\frac{(y+5)(y-1)}{(y+1)(4-y)}$$

The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a. Express the temperature as a single rational expression. b. Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours.

Use a proportion to solve each problem. According to the authors of Number Freaking, in a global village of 200 people, 9 get drunk every day. How many of the world's 6.9 billion people \((2010\) population) get drunk everyday? Round to the nearest hundredth of a billion.
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