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 of 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 finding a common denominator:

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

Find the solutions of 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 factoring the denominators to find the least common multiple:

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

Two 3D printers working together take $6$ hours to manufacture a complete 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 travel 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 entire journey. Set up a rational equation and calculate the initial average speed of the train.

Find the solutions of the following equation, discarding those 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 trip for a total price of $\$400\text{ \$}$. At the last minute, $5$ more students join the trip, which means that each of the original students pays $\$4\text{ \$}$ less. Set up a rational equation to calculate how many students made up 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 the base of the poster were increased by $10\text{ cm}$ and the height reduced 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 the 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 equivalent resistance of the combination, $R_{eq}$, is calculated using the formula $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$. It is known that the value of resistor $R_2$ 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 makes a journey along a river, traveling $24\text{ km}$ 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 factoring the denominators to operate with the least 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 making the change of variable $t = \frac{x^2+1}{x}$:

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

In a laboratory, there are $50\text{ L}$ of a saline solution with an initial concentration of $0.04\text{ kg/L}$. It is desired to evaporate water 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 of 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 posts of heights $x$ and $x+2$ feet are separated by a certain distance on flat ground. Two straight, taut cables are stretched from the top of each post to the base of the opposite post. The point where the cables cross is at a height of $2.4\text{ ft}$ from the ground. Knowing that the height of the crossing $h$ satisfies the geometric relationship $\frac{1}{h} = \frac{1}{h_1} + \frac{1}{h_2}$ (where $h_1$ and $h_2$ are the heights of the posts), set up a rational equation and calculate the height of both posts.