Solving One Variable Linear Equations/Inequalities - Algebra
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What inequality symbol means “at most”?
What inequality symbol means “at most”?
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$\leq$. Less than or equal to represents maximum value.
$\leq$. Less than or equal to represents maximum value.
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What is the solution to the inequality $\frac{x}{4}>-2$?
What is the solution to the inequality $\frac{x}{4}>-2$?
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$x>-8$. Multiply both sides by 4.
$x>-8$. Multiply both sides by 4.
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What is the solution to the inequality $5-2x>1$?
What is the solution to the inequality $5-2x>1$?
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$x<2$. Subtract 5, divide by $-2$, flip sign.
$x<2$. Subtract 5, divide by $-2$, flip sign.
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What is the solution to the inequality $\frac{2x+1}{3}<3$?
What is the solution to the inequality $\frac{2x+1}{3}<3$?
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$x<4$. Multiply by 3, subtract 1, divide by 2.
$x<4$. Multiply by 3, subtract 1, divide by 2.
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What is the solution to the inequality $7\geq 2x-1$?
What is the solution to the inequality $7\geq 2x-1$?
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$x\leq 4$. Add 1, divide by 2, inequality stays same direction.
$x\leq 4$. Add 1, divide by 2, inequality stays same direction.
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What is the solution for $x$ in the literal equation $ax=b$ with $a\neq 0$?
What is the solution for $x$ in the literal equation $ax=b$ with $a\neq 0$?
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$x=\frac{b}{a}$. Divide both sides by $a$ when $a \neq 0$.
$x=\frac{b}{a}$. Divide both sides by $a$ when $a \neq 0$.
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What is the solution for $x$ in the literal equation $ax+b=c$ with $a\neq 0$?
What is the solution for $x$ in the literal equation $ax+b=c$ with $a\neq 0$?
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$x=\frac{c-b}{a}$. Subtract $b$, then divide by $a$.
$x=\frac{c-b}{a}$. Subtract $b$, then divide by $a$.
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What is the solution for $x$ in the literal equation $a(x-b)=c$ with $a\neq 0$?
What is the solution for $x$ in the literal equation $a(x-b)=c$ with $a\neq 0$?
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$x=\frac{c}{a}+b$. Divide by $a$, then add $b$.
$x=\frac{c}{a}+b$. Divide by $a$, then add $b$.
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What is the solution for $x$ in the literal equation $\frac{x}{a}+b=c$ with $a\neq 0$?
What is the solution for $x$ in the literal equation $\frac{x}{a}+b=c$ with $a\neq 0$?
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$x=a(c-b)$. Subtract $b$, then multiply by $a$.
$x=a(c-b)$. Subtract $b$, then multiply by $a$.
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What does it mean to solve an equation in one variable?
What does it mean to solve an equation in one variable?
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Find all values that make the equation true. This is the definition of solving equations.
Find all values that make the equation true. This is the definition of solving equations.
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What property justifies adding the same number to both sides of an equation?
What property justifies adding the same number to both sides of an equation?
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Addition Property of Equality. Maintains equality when adding same value to both sides.
Addition Property of Equality. Maintains equality when adding same value to both sides.
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What property justifies subtracting the same number from both sides of an equation?
What property justifies subtracting the same number from both sides of an equation?
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Subtraction Property of Equality. Maintains equality when subtracting same value from both sides.
Subtraction Property of Equality. Maintains equality when subtracting same value from both sides.
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What property justifies multiplying both sides of an equation by the same nonzero number?
What property justifies multiplying both sides of an equation by the same nonzero number?
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Multiplication Property of Equality. Maintains equality when multiplying both sides by same nonzero value.
Multiplication Property of Equality. Maintains equality when multiplying both sides by same nonzero value.
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What property justifies dividing both sides of an equation by the same nonzero number?
What property justifies dividing both sides of an equation by the same nonzero number?
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Division Property of Equality. Maintains equality when dividing both sides by same nonzero value.
Division Property of Equality. Maintains equality when dividing both sides by same nonzero value.
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What is the distributive property written in general form?
What is the distributive property written in general form?
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$a(b+c)=ab+ac$. Multiplying factor distributes across addition inside parentheses.
$a(b+c)=ab+ac$. Multiplying factor distributes across addition inside parentheses.
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What is the first step to solve most linear equations in one variable?
What is the first step to solve most linear equations in one variable?
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Simplify both sides (combine like terms, distribute). Reduces complexity before applying properties of equality.
Simplify both sides (combine like terms, distribute). Reduces complexity before applying properties of equality.
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What is the solution for $x$ in the literal equation $ax+b=dx+e$ with $a\neq d$?
What is the solution for $x$ in the literal equation $ax+b=dx+e$ with $a\neq d$?
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$x=\frac{e-b}{a-d}$. Collect $x$ terms on one side, solve.
$x=\frac{e-b}{a-d}$. Collect $x$ terms on one side, solve.
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What is the solution to the equation $0.5x+4=9$?
What is the solution to the equation $0.5x+4=9$?
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$x=10$. Subtract 4, then divide by 0.5.
$x=10$. Subtract 4, then divide by 0.5.
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What is the solution for $x$ in the literal equation $\frac{x-b}{a}=c$ with $a\neq 0$?
What is the solution for $x$ in the literal equation $\frac{x-b}{a}=c$ with $a\neq 0$?
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$x=ac+b$. Multiply by $a$, then add $b$.
$x=ac+b$. Multiply by $a$, then add $b$.
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What is the definition of a solution set for an equation?
What is the definition of a solution set for an equation?
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The set of all values that satisfy the equation. Contains all values making the equation true.
The set of all values that satisfy the equation. Contains all values making the equation true.
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What does it mean if solving an equation leads to a true statement like $0=0$?
What does it mean if solving an equation leads to a true statement like $0=0$?
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Infinitely many solutions. Every value of the variable satisfies the equation.
Infinitely many solutions. Every value of the variable satisfies the equation.
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What does it mean if solving an equation leads to a false statement like $0=5$?
What does it mean if solving an equation leads to a false statement like $0=5$?
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No solution. No value of the variable satisfies the equation.
No solution. No value of the variable satisfies the equation.
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What is the rule for solving $x+a=b$ for $x$?
What is the rule for solving $x+a=b$ for $x$?
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$x=b-a$. Subtract $a$ from both sides to isolate $x$.
$x=b-a$. Subtract $a$ from both sides to isolate $x$.
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What is the rule for solving $ax=b$ for $x$ when $a\neq 0$?
What is the rule for solving $ax=b$ for $x$ when $a\neq 0$?
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$x=\frac{b}{a}$. Divide both sides by $a$ to isolate $x$.
$x=\frac{b}{a}$. Divide both sides by $a$ to isolate $x$.
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What is the solution for $x$ in the literal equation $p- qx=r$ with $q\neq 0$?
What is the solution for $x$ in the literal equation $p- qx=r$ with $q\neq 0$?
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$x=\frac{p-r}{q}$. Subtract $r$, divide by $q$.
$x=\frac{p-r}{q}$. Subtract $r$, divide by $q$.
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What is the solution for $x$ in the literal equation $m(x+n)=p$ with $m\neq 0$?
What is the solution for $x$ in the literal equation $m(x+n)=p$ with $m\neq 0$?
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$x=\frac{p}{m}-n$. Divide by $m$, then subtract $n$.
$x=\frac{p}{m}-n$. Divide by $m$, then subtract $n$.
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What is the rule for solving $\frac{x}{a}=b$ for $x$ when $a\neq 0$?
What is the rule for solving $\frac{x}{a}=b$ for $x$ when $a\neq 0$?
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$x=ab$. Multiply both sides by $a$ to isolate $x$.
$x=ab$. Multiply both sides by $a$ to isolate $x$.
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What is the rule for clearing denominators in $\frac{x}{a}+\frac{y}{b}=c$?
What is the rule for clearing denominators in $\frac{x}{a}+\frac{y}{b}=c$?
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Multiply both sides by $\text{LCM}(a,b)$. LCM eliminates all denominators simultaneously.
Multiply both sides by $\text{LCM}(a,b)$. LCM eliminates all denominators simultaneously.
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What inequality symbol means “at most”?
What inequality symbol means “at most”?
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$\leq$. Less than or equal to represents maximum value.
$\leq$. Less than or equal to represents maximum value.
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What inequality symbol means “at least”?
What inequality symbol means “at least”?
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$\geq$. Greater than or equal to represents minimum value.
$\geq$. Greater than or equal to represents minimum value.
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