Find constants $A$ and $B$ such that \[\frac{x + 7}{x^2 - x - 2} = \frac{A}{x - 2} + \frac{B}{x + 1}\] for all $x$ such that $x\neq -1$ and $x\neq 2$. Give your answer as the ordered pair $(A, B)$. and Find all values of $t$ such that $\lfloor t\rfloor = 2t + 3$. If you find more than one value, then list the values you find in increasing order, separated by commas.

1.

2. The values of t are: -3, -1

Step-by-step explanation:

Given

Required

Solve for the unknown

Solving

Take LCM

Expand the denominator

Both denominators are equal; So, they can cancel out

Expand the expression on the right hand side

Collect and Group Like Terms

By Direct comparison of the left hand side with the right hand side

Divide both sides by x in

Make A the subject of formula

Substitute 1 - B for A in

Subtract 1 from both sides

Divide both sides by -3

Substitute -2 for B in

Hence;

Solving

Because we're dealing with an absolute function; the possible expressions that can be derived from the above expression are;

   and  

Solving

Make t the subject of formula

Multiply both sides by -1

Solving

Make t the subject of formula

Divide both sides by -3

Hence, the values of t are: -3, -1