GED Mathematical Reasoning
Inequalities
An inequality is an equation with a range of answers instead of one. x > 5 means x is any number bigger than 5. You solve it exactly like an equation, with one single extra rule that trips up almost everybody.
The words first
- greater than >
- bigger than, not including
- at least ≥
- this much or more — including
- at most ≤
- this much or less — including
- solution set
- all the numbers that make it true
The method
SOLVE IT JUST LIKE AN EQUATION
2x + 3 < 11
2x < 8
x < 4
THE ONE EXTRA RULE
Multiply or divide by a NEGATIVE → FLIP the sign.
−2x < 8
−2x ÷ (−2) > 8 ÷ (−2) ← the < became >
x > −4
Why: 3 < 5 is true. Multiply both by −1: −3 and −5.
Is −3 < −5? No. −3 is BIGGER. The direction had to flip.
TRANSLATING WORDS
at least 18 → x ≥ 18
no more than 40 → x ≤ 40
more than 100 → x > 100
Worked all the way through
The problem
Solve −3x + 5 ≥ 20.
- Step 1 — Undo the +5 first. Subtract 5 from both sides.−3x + 5 − 5 ≥ 20 − 5 → −3x ≥ 15
- Step 2 — Now divide both sides by −3.Notice you are dividing by a negative number.
- Step 3 — Because it is negative, flip the direction.The ≥ becomes ≤.
- Step 4 — Do the division.−3x ÷ (−3) ≤ 15 ÷ (−3) → x ≤ −5
- Step 5 — Test one number to be sure.Try x = −6: −3(−6) + 5 = 18 + 5 = 23, and 23 ≥ 20 ✓
On the board
−3x + 5 ≥ 20
the problem
−3x + 5 − 5 ≥ 20 − 5
take 5 off BOTH sides
−3x ≥ 15
−3x ÷ (−3) ≤ 15 ÷ (−3)
dividing by a NEGATIVE flips ≥ to ≤
x ≤ −5
Answer
x ≤ −5
A second one, in a different form
Solve 4x − 3 > 9.
4x − 3 > 9
the problem
4x > 12
add 3 to both sides
x > 3
divide by positive 4 — no flip
Answer
x > 3
The usual mistake
Flipping the sign when you only added or subtracted a negative. The flip applies to multiplying and dividing only.
Check yourself
Only flip when you multiply or divide by a negative. Subtracting a negative or adding a negative does NOT flip anything.
Practice it
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