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:

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

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 rents a bus for a field trip for a total price of $8\,000\text{ \$}$. At the last minute, $5$ more students sign up for the trip, which means that each of those who were originally going pays $80\text{ \$}$ 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:

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

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 $x+2$ meters 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.