Creating/Solving One Variable Equations/Inequalities - Algebra
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What is the form of a simple rational function in one variable $x$ with a constant numerator?
What is the form of a simple rational function in one variable $x$ with a constant numerator?
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$y=\frac{a}{x}$. Inverse relationship with constant numerator and variable denominator.
$y=\frac{a}{x}$. Inverse relationship with constant numerator and variable denominator.
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What is the slope-intercept form of a linear equation in one variable $y$ versus $x$?
What is the slope-intercept form of a linear equation in one variable $y$ versus $x$?
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$y=mx+b$. Standard form where $m$ is slope and $b$ is y-intercept.
$y=mx+b$. Standard form where $m$ is slope and $b$ is y-intercept.
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What is the standard form of a linear equation in $x$ and $y$?
What is the standard form of a linear equation in $x$ and $y$?
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$Ax+By=C$. General linear form with integer coefficients $A$, $B$, $C$.
$Ax+By=C$. General linear form with integer coefficients $A$, $B$, $C$.
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What is the vertex form of a quadratic function in one variable $x$?
What is the vertex form of a quadratic function in one variable $x$?
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$y=a(x-h)^2+k$. Shows vertex at $(h,k)$ and vertical stretch/compression $a$.
$y=a(x-h)^2+k$. Shows vertex at $(h,k)$ and vertical stretch/compression $a$.
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What is the standard form of a quadratic equation in one variable $x$?
What is the standard form of a quadratic equation in one variable $x$?
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$ax^2+bx+c=0$. Standard quadratic with coefficients $a$, $b$, $c$ where $a\neq 0$.
$ax^2+bx+c=0$. Standard quadratic with coefficients $a$, $b$, $c$ where $a\neq 0$.
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What is the quadratic formula for solutions of $ax^2+bx+c=0$?
What is the quadratic formula for solutions of $ax^2+bx+c=0$?
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$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived from completing the square on $ax^2+bx+c=0$.
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Derived from completing the square on $ax^2+bx+c=0$.
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What does $\Delta>0$ imply about the solutions of $ax^2+bx+c=0$?
What does $\Delta>0$ imply about the solutions of $ax^2+bx+c=0$?
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$2$ distinct real solutions. Positive discriminant means two x-intercepts.
$2$ distinct real solutions. Positive discriminant means two x-intercepts.
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What does $\Delta=0$ imply about the solutions of $ax^2+bx+c=0$?
What does $\Delta=0$ imply about the solutions of $ax^2+bx+c=0$?
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$1$ real double solution. Zero discriminant means one repeated root (vertex on x-axis).
$1$ real double solution. Zero discriminant means one repeated root (vertex on x-axis).
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What does $\Delta<0$ imply about the solutions of $ax^2+bx+c=0$?
What does $\Delta<0$ imply about the solutions of $ax^2+bx+c=0$?
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No real solutions. Negative discriminant means parabola doesn't cross x-axis.
No real solutions. Negative discriminant means parabola doesn't cross x-axis.
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What is the exponent form of a simple exponential function in one variable $x$?
What is the exponent form of a simple exponential function in one variable $x$?
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$y=a\cdot b^x$. Initial value $a$ with base $b$ raised to power $x$.
$y=a\cdot b^x$. Initial value $a$ with base $b$ raised to power $x$.
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What is the rule for solving $ax=b$ when $a\neq 0$?
What is the rule for solving $ax=b$ when $a\neq 0$?
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$x=\frac{b}{a}$. Divide both sides by coefficient $a$ to isolate $x$.
$x=\frac{b}{a}$. Divide both sides by coefficient $a$ to isolate $x$.
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What happens to an inequality when you multiply both sides by a negative number?
What happens to an inequality when you multiply both sides by a negative number?
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Reverse the inequality sign. Multiplying by negative changes direction of inequality.
Reverse the inequality sign. Multiplying by negative changes direction of inequality.
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What inequality symbol represents “at least”?
What inequality symbol represents “at least”?
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$\geq$. "At least" means greater than or equal to.
$\geq$. "At least" means greater than or equal to.
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What inequality symbol represents “at most”?
What inequality symbol represents “at most”?
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$\leq$. "At most" means less than or equal to.
$\leq$. "At most" means less than or equal to.
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What inequality symbol represents “greater than”?
What inequality symbol represents “greater than”?
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$>$. Strict inequality excludes the boundary value.
$>$. Strict inequality excludes the boundary value.
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What inequality symbol represents “less than”?
What inequality symbol represents “less than”?
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$<$. Strict inequality excludes the boundary value.
$<$. Strict inequality excludes the boundary value.
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Identify the property used to rewrite $x+7=12$ as $x=12-7$.
Identify the property used to rewrite $x+7=12$ as $x=12-7$.
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Subtraction property of equality. Subtract same value from both sides to maintain equality.
Subtraction property of equality. Subtract same value from both sides to maintain equality.
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Identify the property used to rewrite $\frac{x}{5}=3$ as $x=15$.
Identify the property used to rewrite $\frac{x}{5}=3$ as $x=15$.
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Multiplication property of equality. Multiply both sides by reciprocal to isolate variable.
Multiplication property of equality. Multiply both sides by reciprocal to isolate variable.
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What is the zero-product property for solving equations like $(x-2)(x+5)=0$?
What is the zero-product property for solving equations like $(x-2)(x+5)=0$?
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If $ab=0$, then $a=0$ or $b=0$. If product equals zero, at least one factor must be zero.
If $ab=0$, then $a=0$ or $b=0$. If product equals zero, at least one factor must be zero.
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What is the definition of a solution to an equation in one variable?
What is the definition of a solution to an equation in one variable?
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A value that makes the equation true. Substituting solution makes left side equal right side.
A value that makes the equation true. Substituting solution makes left side equal right side.
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What is the definition of a solution to an inequality in one variable?
What is the definition of a solution to an inequality in one variable?
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A value that makes the inequality true. Substituting solution satisfies the inequality condition.
A value that makes the inequality true. Substituting solution satisfies the inequality condition.
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Find $x$ if $3x-5=16$.
Find $x$ if $3x-5=16$.
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$x=7$. Add 5, then divide by 3: $3x=21$, so $x=7$.
$x=7$. Add 5, then divide by 3: $3x=21$, so $x=7$.
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Find $x$ if $\frac{x}{4}+2=9$.
Find $x$ if $\frac{x}{4}+2=9$.
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$x=28$. Subtract 2, then multiply by 4: $\frac{x}{4}=7$, so $x=28$.
$x=28$. Subtract 2, then multiply by 4: $\frac{x}{4}=7$, so $x=28$.
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Find $x$ if $5(x-3)=20$.
Find $x$ if $5(x-3)=20$.
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$x=7$. Distribute: $5x-15=20$, then $5x=35$, so $x=7$.
$x=7$. Distribute: $5x-15=20$, then $5x=35$, so $x=7$.
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Find $x$ if $2x+3=3x-5$.
Find $x$ if $2x+3=3x-5$.
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$x=8$. Move variables to one side: $-x=-8$, so $x=8$.
$x=8$. Move variables to one side: $-x=-8$, so $x=8$.
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Solve the inequality $2x+1\leq 9$.
Solve the inequality $2x+1\leq 9$.
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$x\leq 4$. Subtract 1, then divide by 2: $2x\leq 8$.
$x\leq 4$. Subtract 1, then divide by 2: $2x\leq 8$.
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Solve the inequality $-3x>12$.
Solve the inequality $-3x>12$.
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$x<-4$. Divide by $-3$ and flip inequality sign.
$x<-4$. Divide by $-3$ and flip inequality sign.
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Solve the inequality $\frac{x}{-2}\geq 5$.
Solve the inequality $\frac{x}{-2}\geq 5$.
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$x\leq -10$. Multiply by $-2$ and flip inequality sign.
$x\leq -10$. Multiply by $-2$ and flip inequality sign.
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Solve the inequality $4-2x<10$.
Solve the inequality $4-2x<10$.
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$x>-3$. Subtract 4, divide by $-2$, and flip inequality.
$x>-3$. Subtract 4, divide by $-2$, and flip inequality.
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Find the solution set of $|x-3|=5$.
Find the solution set of $|x-3|=5$.
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$x=8$ or $x=-2$. Distance from 3 is 5, so $x-3=5$ or $x-3=-5$.
$x=8$ or $x=-2$. Distance from 3 is 5, so $x-3=5$ or $x-3=-5$.
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