Creating/Solving One Variable Equations/Inequalities - Algebra 2

1. What is the solution to the absolute value equation $|2x+1|=7$?

   $x=3$ or $x=-4$. Solve $2x+1=7$ and $2x+1=-7$ separately.

2. What value must be excluded from solutions of $\frac{x+5}{x^2-9}=1$?

   $x\ne 3$ and $x\ne -3$. Denominator $x^2-9=(x-3)(x+3)$ cannot equal zero.

3. What is the solution to $\frac{x}{x-1}=2$, with restrictions stated?

   $x=2$, with $x\ne 1$. Multiply by $(x-1)$ to get $x=2(x-1)$.

4. What is the axis of symmetry for $y=a(x-h)^2+k$?

   $x=h$. Vertical line through the vertex of any parabola.

5. What are the solutions to the quadratic equation $x^2-6x+9=0$?

   $x=3$. Perfect square trinomial $(x-3)^2=0$ gives double root.

6. What is the solution to the linear equation $\frac{x}{4}+3=8$?

   $x=20$. Subtract $3$, then multiply both sides by $4$.

7. What is the interval notation for the solution set of $x< -2$?

   $(-\infty,-2)$. Parenthesis excludes $-2$, extends from negative infinity.

8. What is the key restriction to state when solving $\frac{1}{x-3}=5$?

   $x\ne 3$. Denominator cannot equal zero in rational equations.

9. What is the solution set for the inequality $5-2x\le 9$ in interval notation?

   $[-2,\infty)$. Subtract $5$, divide by $-2$, flip inequality sign.

10. What inequality models "no more than $12$ tickets" if $t$ is the number of tickets?

    $t\le 12$. "No more than" means less than or equal to.

11. What equation models perimeter $24$ of a square with side length $s$?

    $4s=24$. Square perimeter is $4$ times the side length.

12. What is the standard approach to solve $|x-4|<9$?

    Write $-9<x-4<9$. Absolute value less than positive creates compound inequality.

13. What quadratic equation models "a rectangle has area $30$ with length $x+2$ and width $x-3$"?

    $(x+2)(x-3)=30$. Rectangle area equals length times width.

14. What is the standard approach to solve $|x-4|=9$?

    Solve $x-4=9$ and $x-4=-9$. Absolute value equals positive creates two linear equations.

15. What are the solutions to $x^2-5x=0$?

    $x=0$ or $x=5$. Factor as $x(x-5)=0$ using zero product property.

16. What is the vertex $x$-coordinate of $y=x^2-8x+1$?

    $x=4$. Use $x=-\frac{b}{2a}=-\frac{(-8)}{2(1)}=4$.

17. What is a simple rational equation form that requires excluding values making the denominator $0$?

    $\frac{p(x)}{q(x)}=k$ with $q(x)\ne 0$. Fraction equals constant with domain restrictions.

18. What is the solution set of $|x-1|\le 4$ in interval notation?

    $[-3,5]$. Distance from $1$ is at most $4$ units.

19. What is the continuous growth formula for amount after time $t$ at rate $r$?

    $A=Pe^{rt}$. Principal $P$ grows continuously at rate $r$ over time $t$.

20. What equation models "the product of a number and $6$ is $42$" using variable $x$?

    $6x=42$. "Product" means multiplication of the number and $6$.

21. What are the solutions to $x^2+3x-10=0$?

    $x=2$ or $x=-5$. Factor as $(x-2)(x+5)=0$ for two solutions.

22. Identify the first step to create an equation from a word problem involving a changing quantity.

    Define a variable for the unknown. Establishes what quantity you're solving for.

23. What is the solution to the inequality $2^{x}\ge 8$?

    $x\ge 3$. Rewrite as $2^x\ge 2^3$, so $x\ge 3$.

24. What equation models "a number decreased by $7$ is $20$" using variable $x$?

    $x-7=20$. "Decreased by" means subtract $7$ from the number.

25. What operation clears denominators in $\frac{2}{x}+1=5$ to create an equivalent equation?

    Multiply both sides by the LCD. Eliminates fractions by multiplying by common denominator.

26. What is the inequality for "at least $50$" dollars if $d$ is the number of dollars?

    $d\ge 50$. "At least" means greater than or equal to.

27. What is the standard form of a quadratic equation used for solving in one variable?

    $ax^2+bx+c=0$. Standard quadratic form where $a\ne 0$ for solving methods.

28. What is the solution to the exponential equation $5\cdot 2^{x}=40$?

    $x=3$. Divide by $5$ to get $2^x=8=2^3$.

29. What is the solution to the quadratic equation $x^2-9=0$?

    $x=\pm 3$. Factor as $(x-3)(x+3)=0$ to get solutions.

30. What does $b^2-4ac=0$ tell you about the real solutions of $ax^2+bx+c=0$?

    One real solution (double root). Zero discriminant means the parabola touches the $x$-axis once.

31. What is the factored form of a quadratic used to create equations from zeros?

    $y=a(x-r_1)(x-r_2)$. Shows zeros $r_1$ and $r_2$ directly for intercept problems.

32. What is the compound interest formula for amount after $t$ years with $n$ compounds per year?

    $A=P\left(1+\frac{r}{n}\right)^{nt}$. Principal $P$, rate $r$, compounds $n$ times, over $t$ years.

33. What condition on $b$ in $y=ab^x$ represents exponential decay?

    $0<b<1$. Base between $0$ and $1$ creates decreasing exponential function.

34. What is the interval notation for the compound inequality $-1<x\le 4$?

    $(-1,4]$. Parenthesis excludes $-1$, bracket includes $4$.

35. What is the interval notation for the solution set of $x\ge 3$?

    $[3,\infty)$. Bracket includes $3$, parenthesis extends to infinity.

36. What is the solution to the exponential equation $2^x=16$?

    $x=4$. Rewrite as $2^x=2^4$, so $x=4$.

37. What is the solution to the compound inequality $2<3x+2\le 11$?

    $0<x\le 3$. Subtract $2$, then divide by $3$ for both parts.

38. What is the slope-intercept form of a linear equation in one variable context (function form)?

    $y=mx+b$. Standard form for linear functions with slope $m$ and $y$-intercept $b$.

39. What does $b^2-4ac<0$ tell you about the real solutions of $ax^2+bx+c=0$?

    No real solutions (two complex). Negative discriminant means the parabola doesn't cross the $x$-axis.

40. What does $b^2-4ac>0$ tell you about the real solutions of $ax^2+bx+c=0$?

    Two distinct real solutions. Positive discriminant means two $x$-intercepts exist.

41. What condition on $b$ in $y=ab^x$ represents exponential growth?

    $b>1$. Base greater than $1$ creates increasing exponential function.

42. What is the discriminant of $2x^2-4x+7=0$?

    $-40$. $b^2-4ac=(-4)^2-4(2)(7)=16-56=-40$.

43. What is the solution to the rational equation $\frac{1}{x-2}=3$?

    $x=\frac{7}{3}$. Multiply both sides by $(x-2)$, then solve.

44. What equation models "a population starts at $500$ and grows by $6\%$ per year for $t$ years"?

    $P=500(1.06)^{t}$. $6\%$ growth means multiply by $1.06$ each year.

45. What is the discriminant of $ax^2+bx+c=0$ that determines the number of real solutions?

    $b^2-4ac$. Expression under the square root in the quadratic formula.

46. What is the quadratic formula for solutions of $ax^2+bx+c=0$?

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derives solutions when factoring or completing the square fails.

47. What is the solution to the linear equation $3x-7=11$?

    $x=6$. Add $7$ to both sides, then divide by $3$.

48. What is the general exponential function form used to model growth or decay in one variable?

    $y=ab^x$. Base $b$ and initial value $a$ for exponential modeling.

49. What is the vertex form of a quadratic function often used when creating equations from a maximum or minimum?

    $y=a(x-h)^2+k$. Shows vertex at $(h,k)$ directly for optimization problems.

50. What inequality symbol results after multiplying both sides of an inequality by a negative number?

    The inequality sign reverses. Multiplying by negative flips the inequality direction.

51. What is the minimum value of $y=(x-2)^2-5$?

    $-5$. Vertex form shows minimum value $k=-5$ directly.

52. What is the solution set for the inequality $-3x>12$ in interval notation?

    $(-\infty,-4)$. Divide by $-3$ and flip the inequality direction.

53. What is the solution to the exponential equation $3^{x}=\frac{1}{9}$?

    $x=-2$. Rewrite as $3^x=3^{-2}$, so $x=-2$.

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