Properties of Equality
This property states that if a=b, then a + c =b + c
Addition Property of Equality
Solve 4−3x>10.
x<−2
Find the domain of sqrt(3x-6).
What is x≥2
A student solves 3(x−2)=3x−6 and concludes the equation has infinitely many solutions.
This is the correct justification for that conclusion.
simplifies to a true statement (0 = 0), meaning it is an identity
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SKIN A CAT *he hates cats*
This property allows us to divide both sides of an equation by the same number, as long as it is not zero.
Division Property of Equality
Solve −2(x+1)≤6
x≥−4
Find the domain of 2/(x^2-9).
x≠−3 and x≠3
A student multiplies both sides of an equation by x−5. This step is not reversible because it may introduce this type of solution.
a solution that makes the original expression undefined (division by zero)
"That's hardcore" *why not an avocado?"
Two equations that share the same solution set are described by this term.
Equivalent Equations
Solve ∣2x−3∣<5.
−1<x<4
Find the domain of ln(x−4).
What is x>4
If solving an equation results in 0=7, this is the correct conclusion about the solution set.
What is the equation has no solution (a contradiction)
"Do you think that a monk was working in his monkery and he decided..."
"I shall write this down with my quill pen"
This property states that if a=b, then ac=bc.
Multiplication Property of Equality
Solve ∣x+2∣>3.
x>1 or x<−5
Solve (x+2)/(x-1)=1 and state the domain restriction.
x can't equal 1 and the solution is x=3
Solving an equation leads to x=2 and x=5, but the original equation contains 1/(x−5).
This is the correct final solution set.
2
"If wishes and thoughts were candies and nuts..."
"we'd all have a Merry Christmas"
This property is violated if we divide by zero while solving an equation.
Division Property of Equality
Explain why multiplying an inequality by a negative reverses the inequality symbol using number line reasoning.
Multiplying by a negative reflects numbers across zero and reverses order
Explain why solving before identifying domain restrictions is mathematically incorrect.
It allows values that make the original expression undefined to appear as solutions
Explain why dividing both sides of an equation by an expression containing a variable is logically dangerous.
It may divide by zero for some values and remove valid solutions or introduce extraneous ones
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