Algebra.

Algebra
Setup equation and inequality

• X is 5 more than y → x = y + 5




er
• X is 2 fewer than y → x = y – 2
• X is 2 more than twice y → x = 2y + 2
• X less than y x<y




lp
• X is at least m x≥m
• X is at most m x≤m
• X is no more than m x ≤ m




He
𝟏
• One - half of x → 𝒙
𝟐
𝟏
• One – third of x → 𝒙
𝟑
• Square root of x → √𝒙
• Square of x → 𝒙𝟐



H
• Square of the sum of xx and y → ( x + y )𝟐
•
•
How many more
The difference
→ subtraction
→ subtraction
AT
• Product of x and y → xy
TM
𝟏
• Reciprocal of x →
𝒙
Difference of two squares
SA


𝒙𝟐 − 𝒚𝟐 = (𝒙 − 𝒚)(𝒙 + 𝒚)
𝒙𝟒 − 𝒚𝟒 = (𝒙𝟐 − 𝒚𝟐 )(𝒙𝟐 + 𝒚𝟐 ) =(𝒙 − 𝒚)(𝒙 + 𝒚)(𝒙𝟐 + 𝒚𝟐 )
𝟏 𝟏 𝟏
𝒙𝟐 − 𝒂 = (𝒙 − √𝒂)(𝒙 + √𝒂) 𝒙𝟐 − = (𝒙 − )(𝒙 + )
𝒂𝟐 𝒂 𝒂
rd




𝒙𝟐 𝒚𝟐 − 𝒎𝟐 = (𝒙𝒚 − 𝒎)(𝒙𝒚 + 𝒎)
𝒙𝟐 − 𝒚𝟐 ≠ (𝒙 − 𝒚)𝟐
oa
rB




AmrBoard SAT Helper - Algebra 1
Am

, er
lp
He
H
AT
TM
SA
rd
oa
rB




AmrBoard SAT Helper - Algebra 2
Am

, Exponents – Powers
1) 𝒙𝒏 = 𝒙𝒎 → 𝒏 = 𝒎 2) 𝒙𝟎 = 𝟏 , 𝒙 ≠ 𝟎
𝒂 𝒂𝒏
3) (𝒂𝒃)𝒙 = 𝒂𝒙 , 𝒃𝒙 4) (𝒃)𝒏 = 𝒃𝟐




er
5) 𝒂𝒏 × 𝒂𝒎 = 𝒂𝒏+𝒎 6)𝒂𝒏 ÷ 𝒂𝒎 = 𝒂𝒏−𝒎
𝒎
𝟏 𝒏 𝒏
7) 𝒂−𝒏 = 𝒂𝒏 8) √𝒂𝒎 = 𝒂 𝒏 = ( √𝒂)𝒎




lp
𝟑
9) (√𝒂)𝟐 = 𝒂 ( √ 𝒂) 𝟑 = 𝒂
𝟑
10) √𝒂𝟐 = 𝒂 ,𝒂 > 𝟎 √𝒂𝟑 = 𝒂




He
Rates:
𝒙𝟏 𝒙𝟐
→ x direct variation with y x∝y → =
𝒚𝟏 𝒚𝟐
𝟏
→ x inversely proportion with y x ∝ 𝒚 → 𝒙 𝟏 𝒚𝟏 = 𝒙 𝟐 𝒚𝟐

→ if amr do a job in 𝒕𝟏 hours & ahmed do the same job in 𝒕𝟐 𝒉𝒐𝒖𝒓𝒔




H
If they work together it takes T hours such that
𝟏
𝒕𝟏
+
𝟏
𝒕𝟐

factorization by grouping: ( 4 – terms )
=
𝟏
𝒕
AT
• 𝒙𝟑 + 𝒙𝟐 + 𝒙 + 𝟏 = 𝟎 • 𝒚𝟑 − 𝟑𝒚𝟐 − 𝟒𝒚 + 𝟏𝟐 = 𝟎
TM
𝒙𝟐 (𝒙𝟐 + 𝟏) + (𝒙 + 𝟏) = 𝟎 𝒚 𝟐 ( 𝒚 − 𝟑 ) − 𝟒 (𝒚 − 𝟑 ) = 𝟎
(𝒙 + 𝟏)(𝒙𝟐 + 𝟏) = 𝟎 (y–3) (𝒚𝟐 − 𝟒) = 0
SA


X+1=0 𝒙𝟐 + 𝟏 = 𝟎 (y–3)(y–2)(y+2)=0
X = -1 no soln. Y=3 y=2 y = -1
• 𝒙𝟓 + 𝒙𝟒 − 𝒙𝟑 − 𝒙𝟐 = 𝟎
𝒙𝟐 (𝒙𝟑 + 𝒙𝟐 − 𝒙 − 𝟏) = 𝟎
rd




𝒙𝟐 [𝒙𝟐 (𝒙 + 𝟏) − (𝒙 + 𝟏)] = 𝟎
𝒙𝟐 [(𝒙 + 𝟏)(𝒙𝟐 − 𝟏)] = 𝟎
𝒙𝟐 ( 𝒙 + 𝟏) ( 𝒙 + 𝟏 ) ( 𝒙 − 𝟏 ) = 𝟎
oa




X=0 x = -1 x=1
(double) ( double)
rB




AmrBoard SAT Helper - Algebra 3
Am