GED Mathematical Reasoning
Simplifying Expressions
An expression is a piece of math with no equals sign — like 3x + 5. Simplifying means writing the same thing more compactly. You are not solving anything; you are tidying up.
The words first
- term
- a chunk separated by + or − signs: 3x, 5, −2y
- coefficient
- the number stuck to the front of a variable — the 3 in 3x
- like terms
- terms with the exact same letter part — 3x and 5x are alike, 3x and 3x² are not
- distribute
- multiply what is outside a parenthesis by everything inside it
The method
COMBINING LIKE TERMS — only add what matches
3x + 5x = 8x ✓ both are x
3x + 5y stays 3x + 5y ✗ different letters, cannot combine
3x + 5x² stays as is ✗ different powers, cannot combine
Think of it as 3 apples + 5 apples = 8 apples,
but 3 apples + 5 oranges is just 3 apples and 5 oranges.
DISTRIBUTING — the outside number hits EVERY term inside
3(x + 4) = 3·x + 3·4 = 3x + 12
A MINUS OUTSIDE FLIPS EVERY SIGN INSIDE
−(x − 5) = −x + 5
−2(x − 5) = −2x + 10
Worked all the way through
The problem
Simplify 4(2x + 3) − 2(x − 5).
- Step 1 — Distribute the 4 into the first parenthesis.4 × 2x = 8x and 4 × 3 = 12, so that piece is 8x + 12.
- Step 2 — Distribute the −2 into the second parenthesis. Watch the signs.−2 × x = −2x, and −2 × (−5) = +10, because negative times negative is positive.
- Step 3 — Write it all out with no parentheses.8x + 12 − 2x + 10
- Step 4 — Group the x terms and the plain numbers.(8x − 2x) + (12 + 10)
- Step 5 — Combine each group.6x + 22
On the board
4(2x + 3) − 2(x − 5)
the problem
8x + 12 − 2x + 10
distribute; −2 × −5 = +10
(8x − 2x) + (12 + 10)
group the like terms
6x + 22
Answer
6x + 22
A second one, in a different form
Simplify 5x − 3(x − 2).
5x − 3(x − 2)
the problem
5x − 3x + 6
−3 × −2 = +6
2x + 6
combine the x terms
Answer
2x + 6
The usual mistake
Distributing to the first term only and forgetting the second.
Check yourself
Nearly every error here is that −2 × (−5). A minus in front of a parenthesis changes the sign of everything inside — write the distribution out on its own line rather than doing it in your head.
Practice it
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