Creating/Solving One Variable Equations/Inequalities - Algebra 1

1. What is the definition of a solution to an inequality in one variable?

   A value that makes the inequality true. Substituting solution satisfies the inequality condition.

2. What is the rule for solving $ax=b$ when $a\neq 0$?

   $x=\frac{b}{a}$. Divide both sides by coefficient $a$ to isolate $x$.

3. Solve by factoring: $x^2-4x-12=0$.

   $x=6$ or $x=-2$. Factor as $(x-6)(x+2)=0$ using zero-product property.

4. Identify the equation for "a number $x$ decreased by $6$ is the same as $2x$."

   $x-6=2x$. "Decreased by" means subtract, "same as" means equals.

5. Find the solution set of $|2x+1|\leq 5$.

   $-3\leq x\leq 2$. Distance condition gives $-5\leq 2x+1\leq 5$.

6. Identify the equation that models "$x$ is $5$ more than $y$" in one variable $x$ in terms of $y$.

   $x=y+5$. "More than" means add 5 to the second variable.

7. Solve using the quadratic formula: $x^2-2x-3=0$.

   $x=3$ or $x=-1$. Apply formula with $a=1$, $b=-2$, $c=-3$.

8. What is the zero-product property for solving equations like $(x-2)(x+5)=0$?

   If $ab=0$, then $a=0$ or $b=0$. If product equals zero, at least one factor must be zero.

9. What is the form of a simple rational function in one variable $x$ with a constant numerator?

   $y=\frac{a}{x}$. Inverse relationship with constant numerator and variable denominator.

10. Solve by square roots: $(x-1)^2=16$.

    $x=5$ or $x=-3$. Take square root: $x-1=\pm 4$.

11. Find $x$ if $\frac{x}{4}+2=9$.

    $x=28$. Subtract 2, then multiply by 4: $\frac{x}{4}=7$, so $x=28$.

12. What is the standard form of a quadratic equation in one variable $x$?

    $ax^2+bx+c=0$. Standard quadratic with coefficients $a$, $b$, $c$ where $a\neq 0$.

13. Create an equation for "a rectangle has perimeter $30$ with length $x$ and width $5$."

    $2x+10=30$. Perimeter formula: $P=2l+2w=2x+2(5)$.

14. Create an inequality for "twice a number $x$ is at least $15$."

    $2x\geq 15$. "Twice" means multiply by 2, "at least" means $\geq$.

15. Identify the excluded value for $\frac{5}{x+3}$.

    $x\neq -3$. Denominator cannot equal zero, so $x+3\neq 0$.

16. Create an equation for "$7$ less than a number $n$ is $20$."

    $n-7=20$. "Less than" means subtract 7 from the number.

17. Solve the rational equation $\frac{x}{x-1}=2$ with $x\neq 1$.

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

18. What is the definition of a solution to an equation in one variable?

    A value that makes the equation true. Substituting solution makes left side equal right side.

19. What inequality symbol represents "at most"?

    $\leq$. "At most" means less than or equal to.

20. Identify the property used to rewrite $x+7=12$ as $x=12-7$.

    Subtraction property of equality. Subtract same value from both sides to maintain equality.

21. What is the slope-intercept form of a linear equation in one variable $y$ versus $x$?

    $y=mx+b$. Standard form where $m$ is slope and $b$ is y-intercept.

22. Find and correct the inequality error: from $-2x<6$ to $x< -3$.

    Correct: $x>-3$. Must reverse inequality when dividing by negative.

23. Solve the inequality $-3x>12$.

    $x<-4$. Divide by $-3$ and flip inequality sign.

24. Solve the exponential equation $3^{x}=\frac{1}{9}$.

    $x=-2$. Since $\frac{1}{9}=3^{-2}$, we need $x=-2$.

25. Create an inequality for "a number $x$ is at most $12$."

    $x\leq 12$. "At most" translates to less than or equal.

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

    $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived from completing the square on $ax^2+bx+c=0$.

27. Find and correct the equation error: from $\frac{x}{3}=4$ to $x=\frac{4}{3}$.

    Correct: $x=12$. Must multiply by 3, not divide by 3.

28. What inequality symbol represents "less than"?

    $<$. Strict inequality excludes the boundary value.

29. Solve the inequality $x^2-4\geq 0$.

    $x\leq -2$ or $x\geq 2$. Parabola opens up with zeros at $x=\pm 2$.

30. Solve the rational equation $\frac{6}{x}=2$ with $x\neq 0$.

    $x=3$. Cross-multiply to get $6=2x$, so $x=3$.

31. Solve by square roots: $x^2=49$.

    $x=7$ or $x=-7$. Take square root of both sides: $x=\pm 7$.

32. Solve the exponential equation $5\cdot 2^x=40$.

    $x=3$. Divide by 5: $2^x=8=2^3$, so $x=3$.

33. Find $x$ if $2x+3=3x-5$.

    $x=8$. Move variables to one side: $-x=-8$, so $x=8$.

34. Solve the inequality $2x+1\leq 9$.

    $x\leq 4$. Subtract 1, then divide by 2: $2x\leq 8$.

35. Solve the inequality $\frac{x}{-2}\geq 5$.

    $x\leq -10$. Multiply by $-2$ and flip inequality sign.

36. Create an equation for "the product of $4$ and $x$ increased by $9$ is $25$."

    $4x+9=25$. "Product increased by" means multiply then add.

37. Find $x$ if $3x-5=16$.

    $x=7$. Add 5, then divide by 3: $3x=21$, so $x=7$.

38. Find the solution set of $|x-3|=5$.

    $x=8$ or $x=-2$. Distance from 3 is 5, so $x-3=5$ or $x-3=-5$.

39. Create a rational equation for "$12$ divided by $x$ equals $3$," with $x\neq 0$.

    $\frac{12}{x}=3$. Direct translation of division statement into equation.

40. Solve the inequality $(x-1)(x+2)<0$.

    $-2<x<1$. Product negative when factors have opposite signs.

41. What inequality symbol represents "at least"?

    $\geq$. "At least" means greater than or equal to.

42. What happens to an inequality when you multiply both sides by a negative number?

    Reverse the inequality sign. Multiplying by negative changes direction of inequality.

43. What inequality symbol represents "greater than"?

    $>$. Strict inequality excludes the boundary value.

44. What does $\Delta<0$ imply about the solutions of $ax^2+bx+c=0$?

    No real solutions. Negative discriminant means parabola doesn't cross x-axis.

45. Solve the inequality $4-2x<10$.

    $x>-3$. Subtract 4, divide by $-2$, and flip inequality.

46. Create a quadratic equation for "the area of a square is $49$" using side length $s$.

    $s^2=49$. Area of square equals side length squared.

47. Identify the property used to rewrite $\frac{x}{5}=3$ as $x=15$.

    Multiplication property of equality. Multiply both sides by reciprocal to isolate variable.

48. Solve by factoring: $x^2-9=0$.

    $x=3$ or $x=-3$. Difference of squares: $(x-3)(x+3)=0$.

49. Find $x$ if $5(x-3)=20$.

    $x=7$. Distribute: $5x-15=20$, then $5x=35$, so $x=7$.

50. What does $\Delta>0$ imply about the solutions of $ax^2+bx+c=0$?

    $2$ distinct real solutions. Positive discriminant means two x-intercepts.

51. What does $\Delta=0$ imply about the solutions of $ax^2+bx+c=0$?

    $1$ real double solution. Zero discriminant means one repeated root (vertex on x-axis).

52. What is the vertex form of a quadratic function in one variable $x$?

    $y=a(x-h)^2+k$. Shows vertex at $(h,k)$ and vertical stretch/compression $a$.

53. Create an exponential equation for "a population starts at $200$ and triples in $t$ periods."

    $P=200\cdot 3^t$. Exponential growth model with initial value and growth factor.

54. Solve the exponential equation $2^x=8$.

    $x=3$. Since $8=2^3$, we need $x=3$.

55. What is the exponent form of a simple exponential function in one variable $x$?

    $y=a\cdot b^x$. Initial value $a$ with base $b$ raised to power $x$.

56. Solve by factoring: $x^2+5x=0$.

    $x=0$ or $x=-5$. Factor out $x$: $x(x+5)=0$.

57. What is the standard form of a linear equation in $x$ and $y$?

    $Ax+By=C$. General linear form with integer coefficients $A$, $B$, $C$.

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