Algebra Flashcards: Solving Rational And Radical Equations

What this deck covers

This deck focuses on Solving Rational And Radical Equations, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Solve $\sqrt{x-1} = \sqrt{5}$.

Answer: $x = 6$. Square both sides: $x - 1 = 5$.

Flashcard 2: What restriction must be placed on $x$ for rac{1}{x-3} to be defined?

Answer: $x \ne 3$. Denominator equals zero when $x = 3$.

Flashcard 3: What is the first step before solving a rational equation with denominators?

Answer: State the domain restrictions: denominators cannot equal $0$. Prevents division by zero in rational expressions.

Flashcard 4: Solve rac{1}{x} + \frac{1}{2} = 1 with $x \ne 0$.

Answer: $x = 2$. Multiply by LCD $2x$: $2 + x = 2x$.

Flashcard 5: Solve $\sqrt{x+4} = x$ over the real numbers.

Answer: $x = 4$. Square both sides, solve quadratic, reject negative solution.

Flashcard 6: Which value is extraneous for $\sqrt{x} + 2 = x$: $x=1$ or $x=4$?

Answer: $x = 1$ is extraneous. Makes left side smaller than right side.

Flashcard 7: What is the least common denominator (LCD) of rac{1}{x^2} and rac{3}{x}?

Answer: $x^2$. Highest power of $x$ needed for both fractions.

Flashcard 8: Solve $\sqrt{x+1} = x$ over the real numbers.

Answer: $x = \frac{1+\sqrt{5}}{2}$. Square both sides, solve quadratic, check solutions.

Flashcard 9: What is the solution to $\sqrt{9-x} = 2$?

Answer: $x = 5$. Square both sides: $9 - x = 4$.

Flashcard 10: Solve rac{2}{x} = \frac{3}{x+1} with $x \ne 0,-1$.

Answer: $x = 2$. Cross-multiply: $2(x+1) = 3x$, so $x = 2$.

Flashcard 11: What is the solution to rac{3}{x+2} = 1 with $x \ne -2$?

Answer: $x = 1$. Cross-multiply: $3 = 1(x+2)$, so $x = 1$.

Flashcard 12: Solve $\sqrt{x-2} = x-4$ over the real numbers.

Answer: $x = 5$. Square both sides, solve quadratic, check validity.

Flashcard 13: What is the solution to $\sqrt{x} = 7$?

Answer: $x = 49$. Square both sides: $x = 49$.

Flashcard 14: What is the solution to rac{2x}{x-5} = 3 with $x \ne 5$?

Answer: $x = 15$. Cross-multiply: $2x = 3(x-5)$, so $x = 15$.

Flashcard 15: Solve $\frac{x}{x} = 1$ with $x \neq 0$.

Answer: All real $x$ with $x \neq 0$. Equation $1 = 1$ is always true for valid $x$.

Flashcard 16: Solve rac{x-1}{x+1} = 0 with $x \ne -1$.

Answer: $x = 1$. Numerator equals zero when $x = 1$.

Flashcard 17: Solve $\sqrt{x-2} = 0$.

Answer: $x = 2$. Square root equals zero when radicand is zero.

Flashcard 18: Which value is extraneous when solving $\sqrt{x-2} = x-4$: $x=3$ or $x=5$?

Answer: $x = 3$ is extraneous. Makes right side negative while left side is positive.

Flashcard 19: Solve $\sqrt{x+4} = 2$.

Answer: $x = 0$. Square both sides: $x + 4 = 4$.

Flashcard 20: What is the solution to $\frac{2x}{x-5} = 3$ with $x \ne 5$?

Answer: $x = 15$. Cross-multiply: $2x = 3(x-5)$, so $x = 15$.

Flashcard 21: Solve $\sqrt{2x-1} = x-1$ over the real numbers.

Answer: $x = 2$. Square both sides, solve quadratic, verify in original.

Flashcard 22: Solve $\sqrt{x-2} = x-4$ over the real numbers.

Answer: $x = 5$. Square both sides, solve quadratic, check validity.

Flashcard 23: Solve rac{1}{x} + \frac{1}{x} = \frac{1}{3} with $x \ne 0$.

Answer: $x = 6$. Combine: $\frac{2}{x} = \frac{1}{3}$, so $6 = x$.

Flashcard 24: Solve $\sqrt{x} + 2 = x$ over the real numbers.

Answer: $x = 4$. Isolate radical, square both sides, check solutions.

Flashcard 25: Which value is extraneous when solving $\sqrt{x+1} = x$: $\frac{1-\sqrt{5}}{2}$ or $\frac{1+\sqrt{5}}{2}$?

Answer: $x = \frac{1-\sqrt{5}}{2}$ is extraneous. Negative value fails the original square root equation.

Flashcard 26: What is the solution to rac{x+1}{x} = 4 with $x \ne 0$?

Answer: $x = \frac{1}{3}$. Cross-multiply: $x+1 = 4x$, so $x = \frac{1}{3}$.

Flashcard 27: Solve $\frac{1}{x-2} = \frac{1}{x+2}$ with $x \neq 2$ and $x \neq -2$.

Answer: No solution. Cross-multiplying gives $x + 2 = x - 2$, impossible.

Flashcard 28: What restriction must be placed on $x$ for $\sqrt{3-2x}$ to be real?

Answer: $3-2x \ge 0$, so $x \le \frac{3}{2}$. Square root requires $3 - 2x \ge 0$.

Flashcard 29: Solve rac{x+4}{x-2} = 1 with $x \ne 2$.

Answer: $x$ has no solution. Clearing denominators leads to $x + 4 = x - 2$.

Flashcard 30: What restriction must be placed on $x$ for $\sqrt{(x+1)(x-5)}$ to be real?

Answer: $(x+1)(x-5) \ge 0$. Product must be non-negative for real square root.

Flashcard 31: What operation can create extraneous solutions when solving rational equations?

Answer: Multiplying both sides by an expression containing the variable. May multiply by zero, creating false solutions.

Flashcard 32: Solve rac{x}{x} = 2 with $x \ne 0$.

Answer: No solution. Simplifies to $1 = 2$, which is false.

Flashcard 33: Solve rac{x-1}{x+1} = 0 with $x \ne -1$.

Answer: $x = 1$. Numerator equals zero when $x = 1$.

Flashcard 34: Which value is extraneous when solving $\sqrt{x-2} = x-4$: $x=3$ or $x=5$?

Answer: $x = 3$ is extraneous. Makes right side negative while left side is positive.

Flashcard 35: What is the solution to $\sqrt{x-5} = 4$?

Answer: $x = 21$. Square both sides: $x - 5 = 16$.

Flashcard 36: Solve $\sqrt{x} = -3$ over the real numbers.

Answer: No real solution. Square roots of real numbers cannot be negative.

Flashcard 37: What restriction must be placed on $x$ for $\sqrt{x-7}$ to be real?

Answer: $x-7 \ge 0$, so $x \ge 7$. Square root requires non-negative radicand.

Flashcard 38: What is an extraneous solution in a rational or radical equation?

Answer: A value that solves the transformed equation but not the original equation. Created by operations that aren't reversible.

Flashcard 39: What is an extraneous solution in a rational or radical equation?

Answer: A value that solves the transformed equation but not the original equation. Created by operations that aren't reversible.

Flashcard 40: What is the solution to rac{x}{x-1} = 2 with $x \ne 1$?

Answer: $x = 2$. Cross-multiply to get $x = 2(x-1)$.

Flashcard 41: What is the least common denominator (LCD) of rac{1}{x^2} and rac{3}{x}?

Answer: $x^2$. Highest power of $x$ needed for both fractions.

Flashcard 42: Solve $\sqrt{x+3} = \sqrt{x-1}$ over the real numbers.

Answer: No solution. Would require $x + 3 = x - 1$, impossible.

Flashcard 43: Solve $\sqrt{x} + 1 = 4$.

Answer: $x = 9$. Isolate radical: $\sqrt{x} = 3$.

Flashcard 44: What is the solution to rac{3}{x+2} = 1 with $x \ne -2$?

Answer: $x = 1$. Cross-multiply: $3 = 1(x+2)$, so $x = 1$.

Flashcard 45: Solve $\sqrt{x+1} + \sqrt{x+1} = 6$.

Answer: $x = 8$. Combine like terms: $2\sqrt{x+1} = 6$.

Flashcard 46: What is the least common denominator (LCD) of rac{1}{x} and rac{1}{x-2}?

Answer: $x(x-2)$. Product of all unique denominators.

Flashcard 47: Solve rac{2}{x} = \frac{3}{x+1} with $x \ne 0,-1$.

Answer: $x = 2$. Cross-multiply: $2(x+1) = 3x$, so $x = 2$.

Flashcard 48: Solve $\sqrt{x-2} = 0$.

Answer: $x = 2$. Square root equals zero when radicand is zero.

Flashcard 49: Solve $\sqrt{x} + 1 = 4$.

Answer: $x = 9$. Isolate radical: $\sqrt{x} = 3$.

Flashcard 50: Solve rac{1}{x} + \frac{1}{2} = 1 with $x \ne 0$.

Answer: $x = 2$. Multiply by LCD $2x$: $2 + x = 2x$.

Flashcard 51: What is the solution to $\sqrt{2x+1} = 5$?

Answer: $x = 12$. Square both sides: $2x + 1 = 25$.

Flashcard 52: What is the first step before solving a rational equation with denominators?

Answer: State the domain restrictions: denominators cannot equal $0$. Prevents division by zero in rational expressions.

Flashcard 53: Identify the best method to eliminate denominators in rac{2}{x} = 5.

Answer: Multiply both sides by $x$ (with restriction $x \ne 0$). Clear the single denominator by multiplying both sides.

Flashcard 54: Which value is extraneous when solving $\sqrt{x+4} = x$: $x=4$ or $x=-1$?

Answer: $x = -1$ is extraneous. Makes $\sqrt{x+4}$ negative, which is impossible.

Flashcard 55: What is the solution to rac{x+1}{x} = 4 with $x \ne 0$?

Answer: $x = \frac{1}{3}$. Cross-multiply: $x+1 = 4x$, so $x = \frac{1}{3}$.

Flashcard 56: What restriction must be placed on $x$ for $\sqrt{3-2x}$ to be real?

Answer: $3-2x \ge 0$, so $x \le \frac{3}{2}$. Square root requires $3 - 2x \ge 0$.

Flashcard 57: What must you always do after clearing denominators or squaring both sides?

Answer: Check each candidate solution in the original equation. Verifies solutions aren't extraneous.

Flashcard 58: What operation can create extraneous solutions when solving rational equations?

Answer: Multiplying both sides by an expression containing the variable. May multiply by zero, creating false solutions.

Flashcard 59: What is the solution to rac{2}{x} = 5 with $x \ne 0$?

Answer: $x = \frac{2}{5}$. Multiply both sides by $x$, then solve $2 = 5x$.

Flashcard 60: Solve $\sqrt{x+3} = \sqrt{x-1}$ over the real numbers.

Answer: No solution. Would require $x + 3 = x - 1$, impossible.

Flashcard 61: Identify the best method to eliminate denominators in $\frac{2}{x} = 5$.

Answer: Multiply both sides by $x$ (with restriction $x \ne 0$). Clear the single denominator by multiplying both sides.

Flashcard 62: What restriction must be placed on $x$ for rac{x+2}{x(x-4)} to be defined?

Answer: $x \ne 0$ and $x \ne 4$. Each factor in denominator cannot equal zero.

Flashcard 63: What restriction must be placed on $x$ for rac{x+2}{x(x-4)} to be defined?

Answer: $x \ne 0$ and $x \ne 4$. Each factor in denominator cannot equal zero.

Flashcard 64: What is the solution to $\sqrt{x-5} = 4$?

Answer: $x = 21$. Square both sides: $x - 5 = 16$.

Flashcard 65: What restriction must be placed on $x$ for rac{5}{2x+1} to be defined?

Answer: $x \ne -\frac{1}{2}$. Denominator equals zero when $2x + 1 = 0$.

Flashcard 66: What is the least common denominator (LCD) of rac{1}{x-1} and rac{1}{x^2-1}?

Answer: $(x-1)(x+1)$. Factor $x^2 - 1 = (x-1)(x+1)$

Flashcard 67: What is the solution to $\frac{1}{x-3} = 2$ with $x \neq 3$?

Answer: $x = \frac{7}{2}$. Cross-multiply: $1 = 2(x-3)$, so $x = \frac{7}{2}$.

Flashcard 68: What is the solution to $\sqrt{x} = 7$?

Answer: $x = 49$. Square both sides: $x = 49$.

Flashcard 69: What restriction must be placed on $x$ for rac{5}{2x+1} to be defined?

Answer: $x \ne -\frac{1}{2}$. Denominator equals zero when $2x + 1 = 0$.

Flashcard 70: Solve rac{x+4}{x-2} = 1 with $x \ne 2$.

Answer: $x$ has no solution. Clearing denominators leads to $x + 4 = x - 2$.

Flashcard 71: Which value is extraneous when solving $\sqrt{2x-1} = x-1$: $x=2$ or $x=-1$?

Answer: $x = -1$ is extraneous. Makes the radical undefined in the original equation.

Flashcard 72: Solve $\sqrt{x+1} + \sqrt{x+1} = 6$.

Answer: $x = 8$. Combine like terms: $2\sqrt{x+1} = 6$.

Flashcard 73: Solve rac{1}{x-1} - \frac{1}{x} = 1 with $x \ne 0,1$.

Answer: $x = \frac{1}{2}$. Use LCD $x(x-1)$ and solve resulting equation.

Flashcard 74: Solve $\sqrt{2x-1} = x-1$ over the real numbers.

Answer: $x = 2$. Square both sides, solve quadratic, verify in original.

Flashcard 75: Which value is extraneous for $\sqrt{x} + 2 = x$: $x=1$ or $x=4$?

Answer: $x = 1$ is extraneous. Makes left side smaller than right side.

Flashcard 76: Which value is extraneous when solving $\sqrt{2x-1} = x-1$: $x=2$ or $x=-1$?

Answer: $x = -1$ is extraneous. Makes the radical undefined in the original equation.

Flashcard 77: What operation commonly creates extraneous solutions in radical equations?

Answer: Squaring both sides of an equation. Can introduce solutions that don't satisfy original equation.