Question 611·Hard·Algebra

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If $\frac{6}{y+2} - 3 = 4 - \frac{2}{y+2}$ , the value of $y$ is between which of the following pairs of numbers?

A rational equation has a variable in a denominator. First rule out any value that makes the denominator zero. Then use a Desmos regression to find the value that makes the two sides equal. If the question asks for a pair of numbers, match the solution to an interval rather than stopping at the decimal Desmos shows.

Hints

Hint 1
A denominator is the bottom of a fraction, and it cannot be zero. What value of $y$ makes $y+2=0$ ? Rule that value out before you solve.
Hint 2
A regression can find the input that makes two sides equal. Replace every $y$ with $x_1$ and change $=$ to $\sim$ . Then compare the value Desmos reports with the endpoints of each interval.

Step-by-step

Solve in Desmos, then locate the interval
Step 1
Rule out division by zero
Both fractions have the same denominator, $y+2$ . A denominator cannot be $0$ , so require:
$y+2\ne0.$
Subtract $2$ to find the forbidden value:
$y\ne-2.$
Step 2
Find the value that makes both sides equal
Type $\frac{6}{x_1+2}-3\sim4-\frac{2}{x_1+2}$ in Desmos. Replace every copy of $y$ with $x_1$ because Desmos uses $y$ for graphing; $\sim$ tells its regression to find the input that makes the sides equal. Under PARAMETERS, Desmos shows $x_1\approx-0.857142857$ . This is not the forbidden value $-2$ .
Step 3
Choose the pair containing the solution
An interval between two numbers contains values greater than its lower endpoint and less than its upper endpoint. Since $-1<-0.857142857<0$ , the value of $y$ is between $-1$ and $0$ . Solve first, then answer what the question asks for: the interval. Choice C.

Question 611·Hard·Algebra

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