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When the equation $2-\frac{3}{b}=\frac{5}{b+2}$ is solved for $b$ , the solutions are $-1$ and $3 .$ Explain why the number line must be separated into five segments by the numbers $-2,-1,0,$ and 3 inorder to check the solution set of the inequality $2-\frac{3}{b}>\frac{5}{b+2}$

Because $-1$ and 3 are solutions of the corresponding equation and $-2$ and 0 are roots of the denominators (and thus values at which the equation is undefined).

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 8

Solving Rational Inequalities

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

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we're told that if the greater than symbol here is replaced with an equal sign Ah, the solution to this equation would be b equals negative one and three. However, if we were to write this inequality out on a number line, we would have that the number line should be split at negative one and three as well as negative two and zero. Well, this is because when b equals negative one or three, then we have that these two sides are equal and we only care about the places where this side is greater than this side. If the left side is equal to the right side than it's not greater than so, we don't care about it in our interval. Additionally, if you plug in B equals two, then this term right here becomes we're sorry. Negative to not positive. We get five over negative two plus two, which is 5/0, which doesn't exist. So we'll have to break the number line there. And this term, if you plug in B equals zero becomes 3/0, which doesn't exist. So the number line will be split up at negative one and three because those will set the two sides equal and it will be split up at negative two and zero because those will make this inequality not exist.

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