Solving Rational and Radical Equations - Algebra 1

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

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

2. What restriction must be placed on $x$ for $\frac{1}{x-3}$ to be defined?

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

3. Solve $\sqrt{x+4} = x$ over the real numbers.

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

4. Solve $\sqrt{x+1} = x$ over the real numbers.

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

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

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

6. Solve $\frac{2}{x} = \frac{3}{x+1}$ with $x \ne 0,-1$.

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

7. What is the solution to $\frac{3}{x+2} = 1$ with $x \ne -2$?

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

8. What is the solution to $\frac{2x}{x-5} = 3$ with $x \ne 5$?

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

9. Solve $\frac{x}{x} = 1$ with $x \neq 0$.

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

10. Solve $\frac{x-1}{x+1} = 0$ with $x \ne -1$.

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

11. Solve $\sqrt{x-2} = 0$.

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

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

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

13. Solve $\sqrt{x+4} = 2$.

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

14. What is the solution to $\frac{2x}{x-5} = 3$ with $x \ne 5$?

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

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

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

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

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

17. Solve $\frac{1}{x} + \frac{1}{x} = \frac{1}{3}$ with $x \ne 0$.

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

18. Solve $\sqrt{x} + 2 = x$ over the real numbers.

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

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

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

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

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

21. Solve $\frac{x+4}{x-2} = 1$ with $x \ne 2$.

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

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

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

23. What operation can create extraneous solutions when solving rational equations?

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

24. Solve $\frac{x}{x} = 2$ with $x \ne 0$.

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

25. What is the solution to $\sqrt{x-5} = 4$?

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

26. Solve $\sqrt{x} = -3$ over the real numbers.

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

27. What restriction must be placed on $x$ for $\sqrt{x-7}$ to be real?

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

28. What is an extraneous solution in a rational or radical equation?

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

29. What is the solution to $\frac{x}{x-1} = 2$ with $x \ne 1$?

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

30. What is the least common denominator (LCD) of $\frac{1}{x^2}$ and $\frac{3}{x}$?

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

31. Solve $\sqrt{x} + 1 = 4$.

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

32. Solve $\sqrt{x+1} + \sqrt{x+1} = 6$.

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

33. What is the least common denominator (LCD) of $\frac{1}{x}$ and $\frac{1}{x-2}$?

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

34. Solve $\frac{1}{x} + \frac{1}{2} = 1$ with $x \ne 0$.

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

35. What is the solution to $\sqrt{2x+1} = 5$?

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

36. What is the first step before solving a rational equation with denominators?

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

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

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

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

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

39. What is the solution to $\frac{x+1}{x} = 4$ with $x \ne 0$?

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

40. What restriction must be placed on $x$ for $\sqrt{3-2x}$ to be real?

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

41. What must you always do after clearing denominators or squaring both sides?

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

42. What is the solution to $\frac{2}{x} = 5$ with $x \ne 0$?

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

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

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

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

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

45. What restriction must be placed on $x$ for $\frac{x+2}{x(x-4)}$ to be defined?

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

46. What is the least common denominator (LCD) of $\frac{1}{x-1}$ and $\frac{1}{x^2-1}$?

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

47. What is the solution to $\frac{1}{x-3} = 2$ with $x \neq 3$?

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

48. What is the solution to $\sqrt{x} = 7$?

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

49. What restriction must be placed on $x$ for $\frac{5}{2x+1}$ to be defined?

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

50. Solve $\frac{1}{x-1} - \frac{1}{x} = 1$ with $x \ne 0,1$.

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

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

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

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

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

53. What operation commonly creates extraneous solutions in radical equations?

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

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