Algebra Flashcards: Creating Solving One Variable Equations Inequalities

What this deck covers

This deck focuses on Creating Solving One Variable 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 definition of a solution to an inequality in one variable?

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

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

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

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

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

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

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

Flashcard 5: What is the definition of a solution to an inequality in one variable?

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

Flashcard 6: Find the solution set of $|2x+1|\leq 5$.

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

Flashcard 7: Identify the equation that models "$x$ is $5$ more than $y$" in one variable $x$ in terms of $y$.

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

Flashcard 8: Solve using the quadratic formula: $x^2-2x-3=0$.

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

Flashcard 9: Solve by square roots: $x^2=49$.

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

Flashcard 10: What is the zero-product property for solving equations like $(x-2)(x+5)=0$?

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

Flashcard 11: What is the form of a simple rational function in one variable $x$ with a constant numerator?

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

Flashcard 12: Solve by square roots: $(x-1)^2=16$.

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

Flashcard 13: Find $x$ if $\frac{x}{4}+2=9$.

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

Flashcard 14: What is the standard form of a quadratic equation in one variable $x$?

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

Flashcard 15: Create an equation for "a rectangle has perimeter $30$ with length $x$ and width $5$."

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

Flashcard 16: Find the solution set of $|x-3|=5$.

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

Flashcard 17: Create an inequality for "twice a number $x$ is at least $15$."

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

Flashcard 18: Identify the excluded value for $\frac{5}{x+3}$.

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

Flashcard 19: Create an equation for "$7$ less than a number $n$ is $20$."

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

Flashcard 20: Solve the rational equation $\frac{x}{x-1}=2$ with $x\neq 1$.

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

Flashcard 21: What is the definition of a solution to an equation in one variable?

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

Flashcard 22: What inequality symbol represents "at most"?

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

Flashcard 23: Create an inequality for "a number $x$ is at most $12$."

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

Flashcard 24: Identify the property used to rewrite $x+7=12$ as $x=12-7$.

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

Flashcard 25: Create an equation for "$7$ less than a number $n$ is $20$."

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

Flashcard 26: What is the slope-intercept form of a linear equation in one variable $y$ versus $x$?

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

Flashcard 27: Find and correct the inequality error: from $-2x<6$ to $x< -3$.

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

Flashcard 28: What is the standard form of a quadratic equation in one variable $x$?

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

Flashcard 29: Solve the inequality $-3x>12$.

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

Flashcard 30: Solve the exponential equation $3^{x}=\frac{1}{9}$.

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

Flashcard 31: Create an inequality for "a number $x$ is at most $12$."

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

Flashcard 32: What is the quadratic formula for solutions of $ax^2+bx+c=0$?

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

Flashcard 33: Find and correct the equation error: from $\frac{x}{3}=4$ to $x=\frac{4}{3}$.

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

Flashcard 34: What inequality symbol represents "less than"?

Answer: $<$. Strict inequality excludes the boundary value.

Flashcard 35: Solve the inequality $x^2-4\geq 0$.

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

Flashcard 36: What is the form of a simple rational function in one variable $x$ with a constant numerator?

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

Flashcard 37: Solve the rational equation $\frac{6}{x}=2$ with $x\neq 0$.

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

Flashcard 38: Solve by square roots: $x^2=49$.

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

Flashcard 39: Solve the exponential equation $5\cdot 2^x=40$.

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

Flashcard 40: Find $x$ if $2x+3=3x-5$.

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

Flashcard 41: Solve the inequality $2x+1\leq 9$.

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

Flashcard 42: Solve the inequality $\frac{x}{-2}\geq 5$.

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

Flashcard 43: Create a quadratic equation for "the area of a square is $49$" using side length $s$.

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

Flashcard 44: Create an equation for "the product of $4$ and $x$ increased by $9$ is $25$."

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

Flashcard 45: Find $x$ if $3x-5=16$.

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

Flashcard 46: Find the solution set of $|x-3|=5$.

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

Flashcard 47: Create a rational equation for "$12$ divided by $x$ equals $3$," with $x\neq 0$.

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

Flashcard 48: Solve by factoring: $x^2+5x=0$.

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

Flashcard 49: Solve the inequality $(x-1)(x+2)<0$.

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

Flashcard 50: What inequality symbol represents "at least"?

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

Flashcard 51: What happens to an inequality when you multiply both sides by a negative number?

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

Flashcard 52: What inequality symbol represents "greater than"?

Answer: $>$. Strict inequality excludes the boundary value.

Flashcard 53: What does $\Delta<0$ imply about the solutions of $ax^2+bx+c=0$?

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

Flashcard 54: Solve the inequality $4-2x<10$.

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

Flashcard 55: What inequality symbol represents "greater than"?

Answer: $>$. Strict inequality excludes the boundary value.

Flashcard 56: Create a quadratic equation for "the area of a square is $49$" using side length $s$.

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

Flashcard 57: Identify the property used to rewrite $\frac{x}{5}=3$ as $x=15$.

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

Flashcard 58: Solve by factoring: $x^2-9=0$.

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

Flashcard 59: Find $x$ if $5(x-3)=20$.

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

Flashcard 60: What does $\Delta>0$ imply about the solutions of $ax^2+bx+c=0$?

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

Flashcard 61: What does $\Delta=0$ imply about the solutions of $ax^2+bx+c=0$?

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

Flashcard 62: What is the vertex form of a quadratic function in one variable $x$?

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

Flashcard 63: Create an exponential equation for "a population starts at $200$ and triples in $t$ periods."

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

Flashcard 64: Find $x$ if $\frac{x}{4}+2=9$.

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

Flashcard 65: Solve the exponential equation $2^x=8$.

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

Flashcard 66: What is the exponent form of a simple exponential function in one variable $x$?

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

Flashcard 67: Solve by factoring: $x^2+5x=0$.

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

Flashcard 68: What is the standard form of a linear equation in $x$ and $y$?

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