Algebra Flashcards: Solving One Variable Linear Equations Inequalities

What this deck covers

This deck focuses on Solving One Variable Linear Equations Inequalities, 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: What is the solution to the equation $2(x+3)=2x+5$?

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

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

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

Flashcard 3: What is the solution to the inequality $7\geq 2x-1$?

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

Flashcard 4: What is the solution for $x$ in the literal equation $m(x+n)=p$ with $m\neq 0$?

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

Flashcard 5: What is the solution to the inequality $\frac{2x+1}{3}<3$?

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

Flashcard 6: What is the solution for $x$ in the literal equation $\frac{x}{a}+b=c$ with $a\neq 0$?

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

Flashcard 7: What is the solution for $x$ in the literal equation $ax+b=c$ with $a\neq 0$?

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

Flashcard 8: What is the solution to the inequality $3(x-1)\leq 6$?

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

Flashcard 9: What is the distributive property written in general form?

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

Flashcard 10: What is the solution to the equation $0.5x+4=9$?

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

Flashcard 11: What is the distributive property written in general form?

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

Flashcard 12: What is the solution to the inequality $-2(x+4)\geq 6$?

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

Flashcard 13: What is the solution to the equation $4(x-1)=4x-4$?

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

Flashcard 14: What property justifies adding the same number to both sides of an equation?

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

Flashcard 15: What inequality symbol means "at most"?

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

Flashcard 16: What is the solution to the equation $0.5x+4=9$?

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

Flashcard 17: What inequality symbol means "at most"?

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

Flashcard 18: What happens to an inequality when you multiply or divide both sides by a negative number?

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

Flashcard 19: What is the solution for $x$ in the literal equation $ax=b$ with $a\neq 0$?

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

Flashcard 20: What is the solution to the equation $3x-7=11$?

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

Flashcard 21: What is the first step to solve most linear equations in one variable?

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

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

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

Flashcard 23: What is the solution for $x$ in the literal equation $a(x-b)=c$ with $a\neq 0$?

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

Flashcard 24: What is the solution to the inequality $x-3<5$?

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

Flashcard 25: What is the rule for clearing denominators in $\frac{x}{a}+\frac{y}{b}=c$?

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

Flashcard 26: What is the solution to the inequality $4-\frac{x}{2}<1$?

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

Flashcard 27: What does it mean to solve an equation in one variable?

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

Flashcard 28: What is the solution to the equation $2x+3=4x-9$?

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

Flashcard 29: What is the solution to the equation $6x+2=2(3x+1)$?

Answer: Infinitely many solutions. Distributive property creates identical expressions.

Flashcard 30: What is the solution to the equation $\frac{x}{2}-\frac{x}{3}=1$?

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

Flashcard 31: What is the solution to the equation $4(x+1)-2x=10$?

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

Flashcard 32: What is the rule for solving $x+a=b$ for $x$?

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

Flashcard 33: What is the solution for $x$ in the literal equation $m(x+n)=p$ with $m\neq 0$?

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

Flashcard 34: What is the definition of a solution set for an equation?

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

Flashcard 35: What is the solution for $x$ in the literal equation $\frac{x-b}{a}=c$ with $a\neq 0$?

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

Flashcard 36: What is the solution to the equation $2(x+3)=2x+5$?

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

Flashcard 37: What is the solution to the equation $7-2x=3x+2$?

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

Flashcard 38: What is the solution to the inequality $\frac{x}{4}>-2$?

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

Flashcard 39: What is the solution to the equation $3x-7=11$?

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

Flashcard 40: What does it mean if solving an equation leads to a true statement like $0=0$?

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

Flashcard 41: What property justifies dividing both sides of an equation by the same nonzero number?

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

Flashcard 42: What is the solution to the inequality $-\frac{1}{2}x\geq 3$?

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

Flashcard 43: What is the solution to the equation $6x+2=2(3x+1)$?

Answer: Infinitely many solutions. Distributive property creates identical expressions.

Flashcard 44: What is the rule for clearing denominators in $\frac{x}{a}+\frac{y}{b}=c$?

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

Flashcard 45: What is the solution to the equation $\frac{2x}{5}=6$?

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

Flashcard 46: What is the solution to the equation $5(x-2)=15$?

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

Flashcard 47: What is the solution for $x$ in the literal equation $p- qx=r$ with $q\neq 0$?

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

Flashcard 48: What does it mean to solve an equation in one variable?

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

Flashcard 49: What is the solution for $x$ in the literal equation $ax+b=dx+e$ with $a\neq d$?

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

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

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

Flashcard 51: What is the solution for $x$ in the literal equation $ax=b$ with $a\neq 0$?

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

Flashcard 52: What is the solution to the equation $7-2x=3x+2$?

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

Flashcard 53: What property justifies dividing both sides of an equation by the same nonzero number?

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

Flashcard 54: What inequality symbol means "at least"?

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

Flashcard 55: What is the solution to the inequality $x-3<5$?

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

Flashcard 56: What is the solution to the equation $5(x-2)=15$?

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

Flashcard 57: What is the solution to the inequality $7\geq 2x-1$?

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

Flashcard 58: What is the solution to the equation $\frac{x}{2}-\frac{x}{3}=1$?

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

Flashcard 59: What is the rule for solving $ax=b$ for $x$ when $a\neq 0$?

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

Flashcard 60: What is the solution to the inequality $-\frac{1}{2}x\geq 3$?

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

Flashcard 61: What is the solution to the inequality $\frac{2x+1}{3}<3$?

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

Flashcard 62: What is the solution to the inequality $2x+1\geq 9$?

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

Flashcard 63: What is the solution to the equation $\frac{2x}{5}=6$?

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

Flashcard 64: What property justifies multiplying both sides of an equation by the same nonzero number?

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

Flashcard 65: What is the solution to the equation $2x+3=4x-9$?

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

Flashcard 66: What is the solution to the inequality $-3x\leq 12$?

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

Flashcard 67: What is the solution to the inequality $\frac{x}{4}>-2$?

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

Flashcard 68: What is the solution to the equation $\frac{x+2}{3}+\frac{x-1}{3}=5$?

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

Flashcard 69: What is the solution to the literal inequality $ax>b$ when $a>0$?

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

Flashcard 70: What is the solution for $x$ in the literal equation $ax+b=dx+e$ with $a\neq d$?

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

Flashcard 71: What is the solution to the equation $2x+5=2x-1$?

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

Flashcard 72: What is the solution to the equation $\frac{x}{3}+2=5$?

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

Flashcard 73: What property justifies adding the same number to both sides of an equation?

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

Flashcard 74: What does it mean if solving an equation leads to a false statement like $0=5$?

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

Flashcard 75: What is the solution to the equation $4(x+1)-2x=10$?

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

Flashcard 76: What property justifies multiplying both sides of an equation by the same nonzero number?

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

Flashcard 77: What is the solution to the inequality $-3x\leq 12$?

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

Flashcard 78: What is the solution to the equation $2x+5=2x-1$?

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

Flashcard 79: What does it mean if solving an equation leads to a false statement like $0=5$?

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

Flashcard 80: What is the first step to solve most linear equations in one variable?

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

Flashcard 81: What is the solution to the inequality $2x+1\geq 9$?

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

Flashcard 82: What is the solution to the equation $\frac{x}{3}+2=5$?

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

Flashcard 83: What property justifies subtracting the same number from both sides of an equation?

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

Flashcard 84: What is the solution for $x$ in the literal equation $p- qx=r$ with $q\neq 0$?

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

Flashcard 85: What is the solution to the equation $\frac{x+2}{3}+\frac{x-1}{3}=5$?

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

Flashcard 86: What is the definition of a solution set for an equation?

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

Flashcard 87: What is the rule for solving $\frac{x}{a}=b$ for $x$ when $a\neq 0$?

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

Flashcard 88: What is the solution for $x$ in the literal equation $ax+b=c$ with $a\neq 0$?

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

Flashcard 89: What is the solution to the equation $2(3x-1)=5x+7$?

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

Flashcard 90: What is the solution to the inequality $3(x-1)\leq 6$?

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

Flashcard 91: What is the solution for $x$ in the literal equation $\frac{x-b}{a}=c$ with $a\neq 0$?

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

Flashcard 92: What is the solution to the equation $\frac{x-1}{4}=3$?

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

Flashcard 93: What is the solution to the inequality $5-2x>1$?

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

Flashcard 94: What happens to an inequality when you multiply or divide both sides by a negative number?

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

Flashcard 95: What is the solution to the equation $9-3(x-2)=0$?

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

Flashcard 96: What is the solution to the equation $2(3x-1)=5x+7$?

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

Flashcard 97: What is the solution to the equation $\frac{x-1}{4}=3$?

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

Flashcard 98: What is the rule for solving $x+a=b$ for $x$?

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

Flashcard 99: What does it mean if solving an equation leads to a true statement like $0=0$?

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

Flashcard 100: What is the solution to the literal inequality $ax>b$ when $a>0$?

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