Rational equations

Solve the following rational equation, clearly indicating its valid solutions:

$\frac{3}{x} - \frac{1}{x+2} = \frac{5}{x(x+2)}$

Find the values of $x$ that satisfy the following equality:

$\frac{x+3}{x-1} = \frac{x-2}{x+1}$

Calculate the real values of $x$ for which the following expression holds:

$\frac{x^2-9}{x-3} = 6$

Find the solutions to the following equation:

$\frac{4}{x-2} - \frac{3}{x+2} = \frac{1}{x^2-4}$

To solve the rational equation:

two students propose different strategies before operating:

• Aoife proposes multiplying each term directly by the product of the two visible denominators, $(x-1)\times(x^2-1)$, in order to avoid factoring.

• Cian insists that $x^2-1 = (x-1)(x+1)$ must be factored first in order to multiply the entire equation only by the least common multiple, $(x-1)(x+1)$.

Compare both approaches: justify what algebraic drawbacks (degree of the resulting equation, calculation volume, and possible appearance of extraneous solutions) Aoife's proposal presents compared to Cian's. Then, solve the equation by applying the optimal method, stating the initial domain restrictions and verifying the validity of the solutions obtained.

Solve the rational equation by first factorising the denominators to find the lowest common multiple:

$\frac{x}{x^2-4} + \frac{1}{x+2} = \frac{3}{x-2}$

Cian has to solve the following rational equation in class:

In his notebook, he writes the following steps:

1. Factor the denominator on the right: $x^2-2x-3 = (x-3)(x+1)$.

2. Multiply the entire equation by $(x-3)(x+1)$ and simplify: $x(x+1) - 3(x-3) = 12$.

3. Expand and collect like terms: $x^2 + x - 3x + 9 = 12 \implies x^2 - 2x - 3 = 0$.

4. Apply the quadratic formula and obtain $x = 3$ and $x = -1$.

5. Conclude: "The equation has two real solutions: $x_1 = 3$ and $x_2 = -1$".

Identify the conceptual error committed by Cian, explain with mathematical rigor why his final conclusion is false by checking what happens when substituting those values into the original equation, and determine what the true solution set is.

Find the solutions to the following equation, discarding any that turn out to be false (extraneous solutions):

$\frac{2x+1}{x^2-x-6} - \frac{x}{x-3} = \frac{1}{x+2}$

Find the values of $x$ that satisfy the following equation:

$\frac{x^2-x}{x^2-4x+3} = \frac{2x}{x-3} - 1$

Two 3D printers working together take $6$ hours to produce a full batch of parts for an engineering project. If they worked separately, printer A would take $5$ hours less than printer B to complete the same batch. Set up a rational equation and determine how long printer A would take working alone.

A freight train must cover a distance of $300\text{ km}$. If it increased its average speed by $10\text{ km/h}$, it would take $1$ hour less to complete the whole journey. Set up a rational equation and calculate the average speed at which the train initially travels.

A group of students hires a bus for an excursion for a total price of $400\text{ \euro}$. At the last minute, $5$ more students sign up for the trip, which means that each of those who were originally going pays $4\text{ \euro}$ less. Set up a rational equation to calculate how many students were in the original group.

A graphic designer is creating a rectangular poster whose area must be exactly $600\text{ cm}^2$. If he increased the base of the poster by $10\text{ cm}$ and reduced the height by $5\text{ cm}$, the total area would remain the same. Set up a rational equation and determine the original dimensions of the poster.

A cycling enthusiast goes on an out-and-back training ride on a flat section of $20\text{ km}$ out and $20\text{ km}$ back ($40\text{ km}$ in total) in County Kildare. According to data from a nearby Met Éireann station, a constant wind of $5\text{ km/h}$ blew in the direction of the road throughout the day, so that on the way out he pedaled with a tailwind and on the way back with a headwind. The cyclist completed the total journey in exactly $3\text{ hours}$, maintaining a constant effort (the same average speed $v$ he would maintain without wind).

1. a) Model the situation by setting up a rational equation where the unknown is $v$ (in $\text{km/h}$), expressing the outbound time and the return time in terms of $v$.
2. b) Determine the domain restrictions: what minimum physical and algebraic condition must the speed $v$ satisfy for the cyclist to be able to return against the wind?
3. c) Solve the equation to find $v$ and calculate separately the exact time he spent on the way out and on the way back.

A motorboat travels along a river, covering $24\text{ km}$ downstream and then returning to the starting point upstream. The total navigation time was $5\text{ hours}$. If the speed of the river current is $2\text{ km/h}$, set up a rational equation to calculate the speed of the boat in still water.

Given the rational equation with parameter $a$:

$\frac{x+a}{x-1} + \frac{x-a}{x+1} = \frac{2x^2+2}{x^2-1}$

Determine for which values of parameter $a$ the equation has infinitely many solutions, and for which values it has no solution.

Solve the following rational equation, first factorising the denominators to operate with the lowest common multiple:

$\frac{1}{x^2-5x+6} + \frac{1}{x^2-7x+12} = \frac{2}{x^2-6x+8}$

Solve the following rational equation by applying the substitution $t = \frac{x^2+1}{x}$:

$\frac{x^2+1}{x} + \frac{6x}{x^2+1} = 5$

Consider the following rational equation depending on the real parameter $k$:

David solves it on the board with this procedure:

1. Factor the denominator $x^2+x-2 = (x-1)(x+2)$ and multiply everything by said least common multiple: $k(x+2) + 2(x-1) = x+5$.

2. Expand and collect terms: $kx + 2k + 2x - 2 = x + 5 \implies (k+1)x = 7 - 2k$.

3. Solve for the unknown: $x = \frac{7-2k}{k+1}$.

David concludes: "For any real value of $k$, the equation always has a unique solution given by that formula".

1. a) Point out the two serious reasoning errors committed by David: an algebraic one when isolating the variable in the presence of a parameter, and another relating to the definition and domain of rational functions.
2. b) Carry out a complete and rigorous discussion in terms of the parameter $k$: specify for which values of $k$ the equation has no solution (explaining the reason in each case), for which ones it has a unique valid solution, and whether there is any value of $k$ for which it has infinitely many solutions.