Level of solution complexity - Solutions - Algebra Inequalities

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Topic	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	Algebra Inequalities
Complexity	1	2	3	4	5	6	7	8	9	10	11					Level of solution complexity 3

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Solve the inequality

\frac{(- 7 \cdot {(\frac{- 2 r + 9}{- r + 3})}^{2} + \frac{- 62 r + 279}{- r + 3} + 20) \cdot (\frac{6 r - 27}{- r + 3} + 2)}{\frac{- 14 r + 63}{- r + 3} + 4} - (3 \cdot {(\frac{- 2 r + 9}{- r + 3})}^{2} + \frac{- 12 r + 54}{- r + 3} + 1) \geq 0

1. Let's find the domain of definition D(f) for the considered expression, taking into account that the division by zero is not defined

{\begin{matrix} \frac{- 14 r + 63}{- r + 3} + 4 \neq 0 \\ - r + 3 \neq 0 \end{matrix}

2. Let's add up fractions applying the rule of addition of fractions

{\begin{matrix} - 18 r + 75 \neq 0 \\ r \neq 3 \end{matrix}

3. Let's solve linear equation(s), applying the addition and multiplication principles of equations equivalence

{\begin{matrix} r \neq \frac{25}{6} \\ r \neq 3 \end{matrix}

4. Let's make a change of the variable (the substitution of Mobius). This substitution is applicable when there are only similar algebraic fractions (such two fractions P/Q and R/T, that P/Q=kR/T, where k is real nonzero number) in the expression

\frac{(- 7 \cdot {(2 y)}^{2} + (62 y) + 20) \cdot ((- 6 y) + 2)}{(14 y) + 4} - (3 \cdot {(2 y)}^{2} + (12 y) + 1) \geq 0

5. Let's raise polynomial(s) to natural power

\frac{(- 7 \cdot (4 y^{2}) + (62 y) + 20) \cdot ((- 6 y) + 2)}{(14 y) + 4} - (3 \cdot (4 y^{2}) + (12 y) + 1) \geq 0

6. Let's multiply polynomials by each other, using the distributive law

\frac{(- 28 y^{2} + (62 y) + 20) \cdot (- 6 y + 2)}{(14 y) + 4} - (12 y^{2} + (12 y) + 1) \geq 0

7. Let's combine polynomials, using the definition of operation of addition

\frac{(- 28 y^{2} + 62 y + 20) \cdot (- 6 y + 2)}{14 y + 4} - (12 y^{2} + 12 y + 1) \geq 0

8. Let's combine polynomials, using the definition of operation of addition

\frac{(- 28 y^{2} + 62 y + 20) \cdot (- 6 y + 2)}{14 y + 4} - (12 y^{2} + 12 y + 1) \geq 0

9. Let's factor quadratic trinomial(s), using the theorem of factorization of quadratic trinomial

\frac{((- 28) \cdot (y - \frac{5}{2}) \cdot (y + \frac{2}{7})) \cdot (- 6 y + 2)}{14 y + 4} - (12 y^{2} + 12 y + 1) \geq 0

10. Applying the main property of fractions, let us reduce fractional expression by the linear nonzero polynomial (expression of the form "ax+b")

(- 2 y + 5) \cdot (- 6 y + 2) - (12 y^{2} + 12 y + 1) \geq 0

11. Let's multiply polynomials by each other, using the distributive law

(12 y^{2} - 4 y - 30 y + 10) - (12 y^{2} + 12 y + 1) \geq 0

12. Let's combine polynomials, using the definition of operation of addition

(12 y^{2} - 4 y - 30 y + 10 - 12 y^{2} - 12 y - 1) \geq 0

13. Let's collect similar terms of polynomial

12 y^{2} - 12 y^{2} - 4 y - 30 y - 12 y + 10 - 1 \geq 0

14. Let's add up coefficients at the like terms

- 46 y + 9 \geq 0

15. Let's multiply both sides of inequality by "-1"

46 y - 9 \leq 0

16. Using the addition principle of equivalence of inequalities, let us move the numeric addend from the left-hand side of the inequality to its right-hand side

46 y \leq 9

17. Using the multiplication principle of equivalence of inequalities, let us divide both sides of the inequality by numerical coefficient at the argument

y \leq \frac{9}{46}

18. Let's make the inverse replacement of function

\frac{r - \frac{9}{2}}{r - 3} \leq \frac{9}{46}

19. Let's move expressions from one side of inequality to the other

\frac{r - \frac{9}{2}}{r - 3} - (\frac{9}{46}) \leq 0

20. Let's add up fractions applying the rule of addition of fractions

\frac{r - \frac{9}{2} - (\frac{9}{46}) \cdot (r - 3)}{r - 3} \leq 0

21. Using the distributive law and the rule of multiplication of polynomials "each by each", let's multiply polynomials in the numerator of the fraction