"For exercise, solve the equations and inequalities. Write..." solved and explained

High School
Algebra
Pre-Algebra
Equations, expressions, and inequalitie

Jaden Easton

Answered question

2021-10-18

For exercise, solve the equations and inequalities. Write the solutions sets to the inequalities in interval notation if possible.

\frac{5}{y - 4} = \frac{3 y}{y + 2} - \frac{2 y^{2} - 14 y}{y^{2} - 2 y - 8}

Answer & Explanation

Alix Ortiz

Skilled2021-10-19Added 109 answers

Step 1 $\frac{5}{y - 4} = \frac{3 y}{y + 2} - \frac{2 y^{2} - 14 y}{y^{2} - 2 y - 8}$
$\Rightarrow \frac{5}{y - 4} = \frac{3 y}{y + 2} - \frac{2 y^{2} - 14 y}{y^{2} - 4 y + 2 y - 8}$
$\Rightarrow \frac{5}{y - 4} = \frac{3 y}{y + 2} - \frac{2 y^{2} - 14 y}{y (y - 4) + 2 (y - 4)}$
$\Rightarrow \frac{5}{y - 4} = \frac{3 y}{y + 2} - \frac{2 y^{2} - 14 y}{(y - 4) (y + 2)}$
$\Rightarrow \frac{5}{y - 4} = \frac{3 y (y - 4) - (2 y^{2} - 14 y)}{(y - 4) (y + 2)}$
$\Rightarrow 5 = \frac{3 y (y - 4) - (2 y^{2} - 14 y)}{(y + 2)}$
$\Rightarrow 5 (y + 2) = 3 y (y - 4) - (2 y^{2} - 14 y)$
$\Rightarrow 5 (y + 2) = 3 y^{2} - 12 y - 2 y^{2} + 14 y$
$\Rightarrow 5 y + 10 = y^{2} + 2 y$
Step 2
$\Rightarrow y^{2} + 2 y - 5 y - 10 = 0$
$\Rightarrow y^{2} - 3 y - 10 = 0$
$\Rightarrow y^{2} - 5 y + 2 y - 10 = 0$
$\Rightarrow y (y - 5) + 2 (y - 5) = 0$
$\Rightarrow (y - 5) (y + 2) = 0$
Either $y = 5 or y = - 2$ is not possible
$∴ y = 5$

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