Equivalent expressions
Many exercises are phrased as “identify the equivalent expression for each of the expressions below”. Equivalent expressions have the same value for every input in their shared domain, even when their algebraic forms look different.
Two expressions \(E_1(x)\) and \(E_2(x)\) are equivalent on a set of inputs if \(E_1(x)=E_2(x)\) for every \(x\) in that set. Rational expressions require attention to restrictions such as \(x \ne a\) when a denominator contains \((x-a)\).
Common sources of equivalence
- Distributive structure. Expressions like \(a(b+c)\) and \(ab+ac\) match for all real numbers.
- Combining like terms. Expressions like \(2x+5x\) and \(7x\) match for all \(x\).
- Factoring and expansion. Expressions like \((x+1)^2\) and \(x^2+2x+1\) match for all \(x\).
- Cancellation with restrictions. Expressions like \(\dfrac{x^2-9}{x-3}\) and \(x+3\) match for \(x \ne 3\), but not at \(x=3\) where the first expression is undefined.
Worked set of equivalences
The expressions below illustrate typical patterns. Equality statements are valid for all real \(x\) unless a restriction is stated.
| Expression | Equivalent expression | Equality statement | Notes |
|---|---|---|---|
| \(3(x-2)+5\) | \(3x-1\) | \(3(x-2)+5 = 3x-6+5 = 3x-1\) | Distributive property and like terms |
| \((x+1)^2\) | \(x^2+2x+1\) | \((x+1)^2 = (x+1)(x+1) = x^2+2x+1\) | Binomial square identity |
| \(\dfrac{x^2-9}{x-3}\) | \(x+3\) | \(\dfrac{x^2-9}{x-3}=\dfrac{(x-3)(x+3)}{x-3}=x+3\) | Restriction: \(x \ne 3\) |
| \(\dfrac{2x}{6}\) | \(\dfrac{x}{3}\) | \(\dfrac{2x}{6}=\dfrac{2}{6}\,x=\dfrac{1}{3}x\) | Reducing common factor |
Shared-domain requirement
Rational expressions highlight why “same values” must be interpreted on a shared domain. For \(x \ne 3\), \[ \frac{x^2-9}{x-3} = x+3. \] At \(x=3\), the left-hand expression is undefined, while the right-hand expression equals \(6\). The simplified form matches everywhere the original expression is defined, but it does not repair a removed domain point.
Common pitfalls
Cancelling terms across addition changes meaning; \(\dfrac{x+3}{x+1}\) does not simplify by cancelling the \(+1\) and \(+3\).
Hidden restrictions occur whenever a denominator is simplified; the restriction comes from the original denominator, not from the simplified result.
Numeric checks can detect non-equivalence but cannot prove equivalence; a proof requires algebraic identities or transformations valid on the intended domain.