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 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 2: 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 3: 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 4: 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 5: 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 6: 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 7: 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 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 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 10: What is the interval notation for the solution set of $x\ge 3$?

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

Flashcard 11: 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 12: 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 13: 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 14: What is the interval notation for the solution set of $x< -2$?

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

Flashcard 15: 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 16: 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 17: 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 18: What is the solution to the exponential equation $3^{x}=\frac{1}{9}$?

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

Flashcard 19: What is the solution to the exponential equation $2^x=16$?

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

Flashcard 20: 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 21: 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 22: 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 23: 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 24: What is the interval notation for the solution set of $x\ge 3$?

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

Flashcard 25: What is the interval notation for the solution set of $x< -2$?

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

Flashcard 26: What is the interval notation for the compound inequality $-1<x\le 4$?

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

Flashcard 27: 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 28: 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 29: 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 30: What is the standard approach to solve $|x-4|<9$?

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