Algebra 2 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 2.

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 absolute value equation $|2x+1|=7$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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