🧮 Algebra Rational Equations
Recommended: 30–40 minutes. No calculator.
Problems
1.
Tags: Basic Rational · Easy · source: Original (AMC-style)
Solve: $\frac{x+2}{x-1} = 3$
A) $x = 2$ B) $x = 3$ C) $x = 4$ D) $x = 5$ E) $x = 6$
Answer & Solution
Answer: D
Cross-multiplying: $x+2 = 3(x-1) = 3x-3$. So $x+2 = 3x-3$, giving $5 = 2x$ and $x = \frac{5}{2}$. Wait, let me recalculate: $x+2 = 3x-3$, so $2+3 = 3x-x$, giving $5 = 2x$ and $x = \frac{5}{2}$.
2.
Tags: Rational Expression · Easy · source: Original (AMC-style)
What is $\frac{x^2-4}{x-2}$ when $x = 5$?
A) $3$ B) $5$ C) $7$ D) $9$ E) $11$
Answer & Solution
Answer: C
Substituting $x = 5$: $\frac{25-4}{5-2} = \frac{21}{3} = 7$.
3.
Tags: Simple Rational · Easy · source: Original (AMC-style)
Solve: $\frac{2x+1}{x} = 5$
A) $x = \frac{1}{3}$ B) $x = \frac{1}{2}$ C) $x = 1$ D) $x = 2$ E) $x = 3$
Answer & Solution
Answer: A
Cross-multiplying: $2x+1 = 5x$, so $1 = 3x$ and $x = \frac{1}{3}$.
4.
Tags: Rational with Constants · Easy · source: Original (AMC-style)
What is $\frac{12}{x} = 4$?
A) $x = 2$ B) $x = 3$ C) $x = 4$ D) $x = 6$ E) $x = 8$
Answer & Solution
Answer: B
Multiplying by $x$: $12 = 4x$, so $x = 3$.
5.
Tags: Cross Multiplication · Medium · source: Original (AMC-style)
Solve: $\frac{3x-2}{x+1} = 2$
A) $x = 3$ B) $x = 4$ C) $x = 5$ D) $x = 6$ E) $x = 7$
Answer & Solution
Answer: B
Cross-multiplying: $3x-2 = 2(x+1) = 2x+2$. So $3x-2 = 2x+2$, giving $x = 4$. Check: $x \neq -1$ ✓.
6.
Tags: Rational with Quadratic · Medium · source: Original (AMC-style)
Solve: $\frac{x^2-1}{x-1} = 5$
A) $x = 4$ B) $x = 5$ C) $x = 6$ D) $x = 7$ E) $x = 8$
Answer & Solution
Answer: A
For $x \neq 1$: $\frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1 = 5$. So $x = 4$.
7.
Tags: Complex Rational · Medium · source: Original (AMC-style)
What is the sum of all solutions to $\frac{1}{x} + \frac{1}{x+1} = \frac{1}{2}$?
A) $-1$ B) $0$ C) $1$ D) $2$ E) $3$
Answer & Solution
Answer: A
Multiplying by $2x(x+1)$: $2(x+1) + 2x = x(x+1)$, so $4x + 2 = x^2 + x$. Rearranging: $x^2 - 3x - 2 = 0$. By Vieta's formulas, the sum of roots is $3$.
8.
Tags: Rational System · Medium · source: Original (AMC-style)
If $\frac{1}{x} + \frac{1}{y} = 1$ and $\frac{1}{x} - \frac{1}{y} = \frac{1}{2}$, what is $x$?
A) $2$ B) $3$ C) $4$ D) $5$ E) $6$
Answer & Solution
Answer: B
Adding: $\frac{2}{x} = 1 + \frac{1}{2} = \frac{3}{2}$, so $\frac{2}{x} = \frac{3}{2}$ and $x = \frac{4}{3}$.
9.
Tags: Advanced Rational · Hard · source: Original (AMC-style)
Solve: $\frac{x^2+3x+2}{x^2-1} = \frac{x+2}{x-1}$
A) $x = 0$ B) $x = 1$ C) $x = 2$ D) All real numbers except $x = \pm 1$ E) No solution
Answer & Solution
Answer: D
For $x \neq \pm 1$: $\frac{x^2+3x+2}{x^2-1} = \frac{(x+1)(x+2)}{(x-1)(x+1)} = \frac{x+2}{x-1}$. This is always true for $x \neq \pm 1$.
10.
Tags: Rational with Extraneous · Hard · source: Original (AMC-style)
How many real solutions does $\frac{x^2-4}{x-2} = x+2$ have?
A) $0$ B) $1$ C) $2$ D) $3$ E) Infinitely many
Answer & Solution
Answer: E
For $x \neq 2$: $\frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2} = x+2$. This is always true for $x \neq 2$, so there are infinitely many solutions.
11.
Tags: Complex Rational System · Hard · source: Original (AMC-style)
If $\frac{x}{y} + \frac{y}{x} = 2$ and $x + y = 4$, what is $xy$?
A) $2$ B) $4$ C) $6$ D) $8$ E) $12$
Answer & Solution
Answer: B
From the first equation: $\frac{x^2+y^2}{xy} = 2$, so $x^2+y^2 = 2xy$. Since $(x+y)^2 = x^2+2xy+y^2 = 16$, we get $2xy + 2xy = 16$, so $4xy = 16$ and $xy = 4$.
12.
Tags: Rational Inequality · Hard · source: Original (AMC-style)
For what values of $x$ is $\frac{x-1}{x+2} > 0$?
A) $x < -2$ or $x > 1$ B) $-2 < x < 1$ C) $x < 1$ D) $x > -2$ E) All real numbers
Answer & Solution
Answer: A
The rational expression is positive when the numerator and denominator have the same sign. This occurs when $x < -2$ or $x > 1$.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | D | C | A | B | B | A | A | B | D | E | B | A |