Solving One Variable Linear Equations/Inequalities - Algebra 1

1. What is the solution to the equation $2(x+3)=2x+5$?

   No solution. Left side gives $2x + 6$, right gives $2x + 5$.

2. What is the solution for $x$ in the literal equation $a(x-b)=c$ with $a\neq 0$?

   $x=\frac{c}{a}+b$. Divide by $a$, then add $b$.

3. What is the solution for $x$ in the literal equation $m(x+n)=p$ with $m\neq 0$?

   $x=\frac{p}{m}-n$. Divide by $m$, then subtract $n$.

4. What is the solution to the inequality $\frac{2x+1}{3}<3$?

   $x<4$. Multiply by 3, subtract 1, divide by 2.

5. What is the solution for $x$ in the literal equation $ax+b=c$ with $a\neq 0$?

   $x=\frac{c-b}{a}$. Subtract $b$, then divide by $a$.

6. What is the distributive property written in general form?

   $a(b+c)=ab+ac$. Multiplying factor distributes across addition inside parentheses.

7. What is the solution to the inequality $-2(x+4)\geq 6$?

   $x\leq -7$. Distribute, add 8, divide by $-2$, flip sign.

8. What property justifies adding the same number to both sides of an equation?

   Addition Property of Equality. Maintains equality when adding same value to both sides.

9. What is the solution to the equation $0.5x+4=9$?

   $x=10$. Subtract 4, then divide by 0.5.

10. What inequality symbol means "at most"?

    $\leq$. Less than or equal to represents maximum value.

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

    The inequality sign reverses. Negative multiplier/divisor flips inequality direction.

12. What is the solution for $x$ in the literal equation $ax=b$ with $a\neq 0$?

    $x=\frac{b}{a}$. Divide both sides by $a$ when $a \neq 0$.

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

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

14. What is the rule for clearing denominators in $\frac{x}{a}+\frac{y}{b}=c$?

    Multiply both sides by $\text{LCM}(a,b)$. LCM eliminates all denominators simultaneously.

15. What is the solution to the equation $2x+3=4x-9$?

    $x=6$. Collect variables on one side, constants on the other.

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

    $x=6$. Find common denominator $\frac{6}{6}$, then solve.

17. What is the solution to the equation $7-2x=3x+2$?

    $x=1$. Collect variables on one side, constants on the other.

18. What is the solution to the inequality $-\frac{1}{2}x\geq 3$?

    $x\leq -6$. Divide by $-\frac{1}{2}$ and flip inequality sign.

19. What is the solution to the equation $6x+2=2(3x+1)$?

    Infinitely many solutions. Distributive property creates identical expressions.

20. What is the solution to the equation $\frac{2x}{5}=6$?

    $x=15$. Multiply both sides by $\frac{5}{2}$.

21. What does it mean to solve an equation in one variable?

    Find all values that make the equation true. This is the definition of solving equations.

22. What is the solution to the literal inequality $ax>b$ when $a<0$?

    $x<\frac{b}{a}$. Negative coefficient reverses inequality direction.

23. What property justifies dividing both sides of an equation by the same nonzero number?

    Division Property of Equality. Maintains equality when dividing both sides by same nonzero value.

24. What inequality symbol means "at least"?

    $\geq$. Greater than or equal to represents minimum value.

25. What is the solution to the inequality $x-3<5$?

    $x<8$. Add 3 to both sides.

26. What is the solution to the equation $5(x-2)=15$?

    $x=5$. Distribute, add 10 to both sides, then divide by 5.

27. What is the solution to the inequality $7\geq 2x-1$?

    $x\leq 4$. Add 1, divide by 2, inequality stays same direction.

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

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

29. What is the solution to the inequality $-3x\leq 12$?

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

30. What is the solution to the inequality $\frac{x}{4}>-2$?

    $x>-8$. Multiply both sides by 4.

31. What is the solution for $x$ in the literal equation $ax+b=dx+e$ with $a\neq d$?

    $x=\frac{e-b}{a-d}$. Collect $x$ terms on one side, solve.

32. What is the solution to the equation $2x+5=2x-1$?

    No solution. Variables cancel, leaving false statement $5=-1$.

33. What is the solution to the equation $\frac{x}{3}+2=5$?

    $x=9$. Subtract 2, then multiply by 3.

34. What does it mean if solving an equation leads to a false statement like $0=5$?

    No solution. No value of the variable satisfies the equation.

35. What is the solution to the equation $4(x+1)-2x=10$?

    $x=3$. Distribute, combine like terms, solve for $x$.

36. What property justifies multiplying both sides of an equation by the same nonzero number?

    Multiplication Property of Equality. Maintains equality when multiplying both sides by same nonzero value.

37. What is the first step to solve most linear equations in one variable?

    Simplify both sides (combine like terms, distribute). Reduces complexity before applying properties of equality.

38. What is the solution to the inequality $2x+1\geq 9$?

    $x\geq 4$. Subtract 1, then divide by 2.

39. What property justifies subtracting the same number from both sides of an equation?

    Subtraction Property of Equality. Maintains equality when subtracting same value from both sides.

40. What is the solution for $x$ in the literal equation $p- qx=r$ with $q\neq 0$?

    $x=\frac{p-r}{q}$. Subtract $r$, divide by $q$.

41. What is the solution to the equation $\frac{x+2}{3}+\frac{x-1}{3}=5$?

    $x=\frac{13}{2}$. Combine fractions with common denominator 3.

42. What is the definition of a solution set for an equation?

    The set of all values that satisfy the equation. Contains all values making the equation true.

43. What is the solution to the equation $2(3x-1)=5x+7$?

    $x=9$. Distribute left side, collect terms, solve for $x$.

44. What is the solution to the inequality $3(x-1)\leq 6$?

    $x\leq 3$. Distribute, add 3, divide by 3.

45. What is the solution for $x$ in the literal equation $\frac{x-b}{a}=c$ with $a\neq 0$?

    $x=ac+b$. Multiply by $a$, then add $b$.

46. What is the solution to the equation $9-3(x-2)=0$?

    $x=5$. Distribute, simplify, then solve for $x$.

47. What is the solution to the equation $\frac{x-1}{4}=3$?

    $x=13$. Add 1, then multiply by 4.

48. What is the rule for solving $x+a=b$ for $x$?

    $x=b-a$. Subtract $a$ from both sides to isolate $x$.

49. What does it mean if solving an equation leads to a true statement like $0=0$?

    Infinitely many solutions. Every value of the variable satisfies the equation.

50. What is the solution to the literal inequality $ax>b$ when $a>0$?

    $x>\frac{b}{a}$. Positive coefficient preserves inequality direction.

51. What is the solution for $x$ in the literal equation $\frac{x}{a}+b=c$ with $a\neq 0$?

    $x=a(c-b)$. Subtract $b$, then multiply by $a$.

52. What is the solution to the inequality $4-\frac{x}{2}<1$?

    $x>6$. Subtract 4, multiply by $-2$, flip sign.

53. What is the solution to the inequality $5-2x>1$?

    $x<2$. Subtract 5, divide by $-2$, flip sign.

54. What is the solution to the equation $4(x-1)=4x-4$?

    Infinitely many solutions. Equation simplifies to true statement $-4=-4$.

55. What is the solution to the inequality $\frac{x-3}{2}\leq 4$?

    $x\leq 11$. Add 3, multiply by 2.

56. What is the rule for solving $\frac{x}{a}=b$ for $x$ when $a\neq 0$?

    $x=ab$. Multiply both sides by $a$ to isolate $x$.

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