How to Solve Rational Equations Step-by-Step

TL;DR

To solve the rational equation 5/(x-1) = 3/(x+3), multiply both sides by the denominators (x+3) and (x-1) to eliminate fractions. This simplifies to 5(x+3) = 3(x-1), which can then be solved for x. The solution is x = -9, provided x does not equal -3 or 1 to avoid division by zero.

Transcript

Solve the equation, 5 over x minus 1 is equal to 3 over x plus 3. And they give us two constraints here. x can't be negative 3 or 1. And that's because if x was negative 3, this expression right here, you'd be dividing by 0, it'd be undefined. If x was 1, this expression right here would be dividing by 0 and it would be undefined. So let's see what... Read More

Key Insights

• ☺️ Constraints of x not being -3 or 1 are necessary to avoid dividing by zero.
• 🙃 Multiplying both sides by the denominators eliminates fractions and simplifies the equation.
• 😵 Cross multiplication is a valid method to remove denominators and solve linear equations.

Questions & Answers

Q: Why can't x be -3 or 1 in this equation?

The equation involves dividing by (x-1) and (x+3), and division by zero is undefined. Therefore, x cannot be -3 or 1.

Q: What is the purpose of multiplying both sides by the denominators?

Multiplying by the denominators eliminates the fractions in the equation, allowing us to work with whole numbers. It simplifies the equation and makes it easier to solve.

Q: How does multiplying by (x-1) and (x+3) cancel out the denominators?

When we multiply (x-1) with (x+3), the (x-1) term cancels out with the denominator (x-1). Similarly, (x+3) cancels out with the denominator (x+3), leaving us with a fraction-free equation.

Q: Why does the equation become 5(x+3) = 3(x-1) after multiplying both sides?

When we multiply both sides by (x+3)(x-1), the denominators cancel out, resulting in cross multiplication. This produces the equation 5(x+3) = 3(x-1), which represents an equivalent form of the original equation.

Summary & Key Takeaways

• The equation given is 5/x-1 = 3/x+3, with the constraints that x can't be -3 or 1.

• To eliminate the denominators, multiply both sides of the equation by (x+3) and (x-1).

• After simplification, the equation becomes 5(x+3) = 3(x-1).

• Solve for x by distributing and rearranging the terms.