Creating/Solving One Variable Equations/Inequalities - Algebra 2
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What value must be excluded from solutions of $\frac{x+5}{x^2-9}=1$?
What value must be excluded from solutions of $\frac{x+5}{x^2-9}=1$?
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$x\ne 3$ and $x\ne -3$. Denominator $x^2-9=(x-3)(x+3)$ cannot equal zero.
$x\ne 3$ and $x\ne -3$. Denominator $x^2-9=(x-3)(x+3)$ cannot equal zero.
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What is the vertex $x$-coordinate of $y=x^2-8x+1$?
What is the vertex $x$-coordinate of $y=x^2-8x+1$?
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$x=4$. Use $x=-\frac{b}{2a}=-\frac{(-8)}{2(1)}=4$.
$x=4$. Use $x=-\frac{b}{2a}=-\frac{(-8)}{2(1)}=4$.
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What is the slope-intercept form of a linear equation in one variable context (function form)?
What is the slope-intercept form of a linear equation in one variable context (function form)?
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$y=mx+b$. Standard form for linear functions with slope $m$ and $y$-intercept $b$.
$y=mx+b$. Standard form for linear functions with slope $m$ and $y$-intercept $b$.
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What condition on $b$ in $y=ab^x$ represents exponential decay?
What condition on $b$ in $y=ab^x$ represents exponential decay?
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$0<b<1$. Base between $0$ and $1$ creates decreasing exponential function.
$0<b<1$. Base between $0$ and $1$ creates decreasing exponential function.
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What does $b^2-4ac=0$ tell you about the real solutions of $ax^2+bx+c=0$?
What does $b^2-4ac=0$ tell you about the real solutions of $ax^2+bx+c=0$?
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One real solution (double root). Zero discriminant means the parabola touches the $x$-axis once.
One real solution (double root). Zero discriminant means the parabola touches the $x$-axis once.
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What is the axis of symmetry for $y=a(x-h)^2+k$?
What is the axis of symmetry for $y=a(x-h)^2+k$?
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$x=h$. Vertical line through the vertex of any parabola.
$x=h$. Vertical line through the vertex of any parabola.
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What is a simple rational equation form that requires excluding values making the denominator $0$?
What is a simple rational equation form that requires excluding values making the denominator $0$?
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$\frac{p(x)}{q(x)}=k$ with $q(x)\ne 0$. Fraction equals constant with domain restrictions.
$\frac{p(x)}{q(x)}=k$ with $q(x)\ne 0$. Fraction equals constant with domain restrictions.
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What is the key restriction to state when solving $\frac{1}{x-3}=5$?
What is the key restriction to state when solving $\frac{1}{x-3}=5$?
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$x\ne 3$. Denominator cannot equal zero in rational equations.
$x\ne 3$. Denominator cannot equal zero in rational equations.
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What is the general exponential function form used to model growth or decay in one variable?
What is the general exponential function form used to model growth or decay in one variable?
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$y=ab^x$. Base $b$ and initial value $a$ for exponential modeling.
$y=ab^x$. Base $b$ and initial value $a$ for exponential modeling.
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What is the interval notation for the solution set of $x\ge 3$?
What is the interval notation for the solution set of $x\ge 3$?
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$[3,\infty)$. Bracket includes $3$, parenthesis extends to infinity.
$[3,\infty)$. Bracket includes $3$, parenthesis extends to infinity.
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What is the vertex $x$-coordinate of $y=x^2-8x+1$?
What is the vertex $x$-coordinate of $y=x^2-8x+1$?
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$x=4$. Use $x=-\frac{b}{2a}=-\frac{(-8)}{2(1)}=4$.
$x=4$. Use $x=-\frac{b}{2a}=-\frac{(-8)}{2(1)}=4$.
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What is the solution set of $|x-1|\le 4$ in interval notation?
What is the solution set of $|x-1|\le 4$ in interval notation?
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$[-3,5]$. Distance from $1$ is at most $4$ units.
$[-3,5]$. Distance from $1$ is at most $4$ units.
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What is the solution to the absolute value equation $|2x+1|=7$?
What is the solution to the absolute value equation $|2x+1|=7$?
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$x=3$ or $x=-4$. Solve $2x+1=7$ and $2x+1=-7$ separately.
$x=3$ or $x=-4$. Solve $2x+1=7$ and $2x+1=-7$ separately.
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What is the interval notation for the solution set of $x< -2$?
What is the interval notation for the solution set of $x< -2$?
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$(-\infty,-2)$. Parenthesis excludes $-2$, extends from negative infinity.
$(-\infty,-2)$. Parenthesis excludes $-2$, extends from negative infinity.
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What does $b^2-4ac<0$ tell you about the real solutions of $ax^2+bx+c=0$?
What does $b^2-4ac<0$ tell you about the real solutions of $ax^2+bx+c=0$?
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No real solutions (two complex). Negative discriminant means the parabola doesn't cross the $x$-axis.
No real solutions (two complex). Negative discriminant means the parabola doesn't cross the $x$-axis.
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What is the solution to the inequality $2^{x}\ge 8$?
What is the solution to the inequality $2^{x}\ge 8$?
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$x\ge 3$. Rewrite as $2^x\ge 2^3$, so $x\ge 3$.
$x\ge 3$. Rewrite as $2^x\ge 2^3$, so $x\ge 3$.
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What is the solution to the exponential equation $5\cdot 2^{x}=40$?
What is the solution to the exponential equation $5\cdot 2^{x}=40$?
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$x=3$. Divide by $5$ to get $2^x=8=2^3$.
$x=3$. Divide by $5$ to get $2^x=8=2^3$.
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What is the solution to the exponential equation $3^{x}=\frac{1}{9}$?
What is the solution to the exponential equation $3^{x}=\frac{1}{9}$?
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$x=-2$. Rewrite as $3^x=3^{-2}$, so $x=-2$.
$x=-2$. Rewrite as $3^x=3^{-2}$, so $x=-2$.
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What is the solution to the exponential equation $2^x=16$?
What is the solution to the exponential equation $2^x=16$?
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$x=4$. Rewrite as $2^x=2^4$, so $x=4$.
$x=4$. Rewrite as $2^x=2^4$, so $x=4$.
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What is the solution to $\frac{x}{x-1}=2$, with restrictions stated?
What is the solution to $\frac{x}{x-1}=2$, with restrictions stated?
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$x=2$, with $x\ne 1$. Multiply by $(x-1)$ to get $x=2(x-1)$.
$x=2$, with $x\ne 1$. Multiply by $(x-1)$ to get $x=2(x-1)$.
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What is the compound interest formula for amount after $t$ years with $n$ compounds per year?
What is the compound interest formula for amount after $t$ years with $n$ compounds per year?
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$A=P\left(1+\frac{r}{n}\right)^{nt}$. Principal $P$, rate $r$, compounds $n$ times, over $t$ years.
$A=P\left(1+\frac{r}{n}\right)^{nt}$. Principal $P$, rate $r$, compounds $n$ times, over $t$ years.
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What is the continuous growth formula for amount after time $t$ at rate $r$?
What is the continuous growth formula for amount after time $t$ at rate $r$?
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$A=Pe^{rt}$. Principal $P$ grows continuously at rate $r$ over time $t$.
$A=Pe^{rt}$. Principal $P$ grows continuously at rate $r$ over time $t$.
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What inequality symbol results after multiplying both sides of an inequality by a negative number?
What inequality symbol results after multiplying both sides of an inequality by a negative number?
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The inequality sign reverses. Multiplying by negative flips the inequality direction.
The inequality sign reverses. Multiplying by negative flips the inequality direction.
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What is the interval notation for the solution set of $x\ge 3$?
What is the interval notation for the solution set of $x\ge 3$?
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$[3,\infty)$. Bracket includes $3$, parenthesis extends to infinity.
$[3,\infty)$. Bracket includes $3$, parenthesis extends to infinity.
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What is the interval notation for the solution set of $x< -2$?
What is the interval notation for the solution set of $x< -2$?
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$(-\infty,-2)$. Parenthesis excludes $-2$, extends from negative infinity.
$(-\infty,-2)$. Parenthesis excludes $-2$, extends from negative infinity.
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What is the interval notation for the compound inequality $-1<x\le 4$?
What is the interval notation for the compound inequality $-1<x\le 4$?
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$(-1,4]$. Parenthesis excludes $-1$, bracket includes $4$.
$(-1,4]$. Parenthesis excludes $-1$, bracket includes $4$.
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Identify the first step to create an equation from a word problem involving a changing quantity.
Identify the first step to create an equation from a word problem involving a changing quantity.
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Define a variable for the unknown. Establishes what quantity you're solving for.
Define a variable for the unknown. Establishes what quantity you're solving for.
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What operation clears denominators in $\frac{2}{x}+1=5$ to create an equivalent equation?
What operation clears denominators in $\frac{2}{x}+1=5$ to create an equivalent equation?
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Multiply both sides by the LCD. Eliminates fractions by multiplying by common denominator.
Multiply both sides by the LCD. Eliminates fractions by multiplying by common denominator.
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What is the standard approach to solve $|x-4|=9$?
What is the standard approach to solve $|x-4|=9$?
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Solve $x-4=9$ and $x-4=-9$. Absolute value equals positive creates two linear equations.
Solve $x-4=9$ and $x-4=-9$. Absolute value equals positive creates two linear equations.
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What is the standard approach to solve $|x-4|<9$?
What is the standard approach to solve $|x-4|<9$?
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Write $-9<x-4<9$. Absolute value less than positive creates compound inequality.
Write $-9<x-4<9$. Absolute value less than positive creates compound inequality.
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