Exercises: 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:

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

Determine the solution to the equation, paying attention to the values that make the denominators zero:

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

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

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

Solve the rational equation by first putting it over a common denominator:

$\frac{1}{x} + \frac{1}{2x} + \frac{1}{3x} = \frac{11}{6}$

Find the solutions to the following equation:

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

Determine the real number $x$ that satisfies the equality between these two algebraic fractions:

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

Solve the following rational equation:

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

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.

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:

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

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.

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

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

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.

Solve the equation by simplifying the algebraic fractions:

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

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.

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

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

After torrential rains caused by a storm in the Shannon basin, firefighters must pump out the water accumulated in an underpass. The usual municipal motor pump would take $6\text{ hours}$ to empty it on its own. By bringing in a second, more modern backup motor pump, both working simultaneously manage to empty it completely in $2.4\text{ hours}$.

1. a) Estimate and reason: Before setting up any equation, justify why the time it would take the modern pump alone must necessarily be less than $6\text{ hours}$ but strictly greater than $2.4\text{ hours}$. Furthermore, explain whether you expect it to take more or less than $3\text{ hours}$ by comparing the $2.4\text{ hours}$ with how long they would take together if the modern pump took exactly $3\text{ hours}$.
2. b) Set up the rational equation for joint work per hour taking $t$ as the unknown (time in hours that the backup pump would take on its own), indicating the domain restrictions of the variable.
3. c) Solve the equation and check whether your exact result matches the estimation you made.

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.

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.

In an electronic circuit, two resistors are connected in parallel. The total equivalent resistance, $R_{eq}$, is calculated using the formula $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$. It is known that the value of resistance $R_2$ is $5\ \Omega$ greater than that of $R_1$, and that the equivalent resistance of the circuit is $6\ \Omega$. Set up and solve a rational equation to find the value of both resistors.

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.

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 $t = \frac{x^2+1}{x}$:

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

In a laboratory, there are $50\text{ L}$ of a saline solution with an initial concentration of $0.04\text{ kg/L}$. Water is to be evaporated from the mixture at a constant rate to increase its concentration (the amount of salt remains unchanged). If $x\text{ L}$ of water is evaporated, the new concentration must be $0.1\text{ kg/L}$. Set up the corresponding rational equation and calculate the amount of water that must be evaporated.

Determine all the real solutions to the equation, simplifying the algebraic fractions using algebraic identities before operating:

$\frac{x^3-8}{x-2} = \frac{x^3+1}{x+1} + 3$

Two vertical poles of heights $x$ and $x+2$ metres are separated by a certain distance on flat ground. Two straight, taut cables are stretched from the top of each pole to the base of the opposite pole. The point where the cables cross is at a height of $2.4\text{ m}$ from the ground. Knowing that the crossing height $h$ satisfies the geometric relation $\frac{1}{h} = \frac{1}{h_1} + \frac{1}{h_2}$ (where $h_1$ and $h_2$ are the heights of the poles), set up a rational equation and calculate the height of both poles.

Reasonably design a rational equation that strictly satisfies each and every one of the following conditions:

1. It must contain at least two algebraic fractions with distinct non-constant denominators.

2. The only values excluded from the domain of the equation must be $x = -2$ and $x = 3$.

3. When applying the standard procedure (multiplying the entire equation by the least common multiple of the denominators to eliminate fractions), the resulting second-degree polynomial equation must have $x = 3$ and $x = 5$ as candidate solutions.

4. When checked against the domain restrictions, the value $x = 3$ must be discarded because it is a false or extraneous solution, leaving $x = 5$ as the only valid solution of the equation.

Show the step-by-step reverse deduction to construct such an equation (starting from the desired roots and reconstructing the algebraic fractions) and formally solve the equation you obtained to prove that it satisfies all the required conditions.

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.