HS Algebra 1

Linear Functions

100

4x + 5 = 21

4x = 16 → x = 4

200

6(x − 1) = 42

x − 1 = 7 → x = 8

300

2x − 9 = 3x + 7

−9 − 7 = x → x = −16

400

Slope of y = 3x − 12

slope = 3

500

(5x/4) − 7 = 18

5x/4 = 25 → 5x = 100 → x = 20

Inequalities

100

x + 3 > 12

x > 9

200

8 − 2x ≤ 0

−2x ≤ −8 → divide by −2 (flip sign) → x ≥ 4

300

3x + 4 < 19

3x < 15 → x < 5

400

−6 ≤ 2x + 2 < 10

Subtract 2 → −8 ≤ 2x < 8
Divide by 2 → −4 ≤ x < 4

500

|x − 5| = 9

Case 1: x − 5 = 9 → x = 14
Case 2: x − 5 = −9 → x = −4
→ x = 14 or x = −4

System Of Equations

100

y = x + 4
x + y = 14

→ substitute → x + (x + 4) = 14
→ 2x = 10 → x = 5 → y = 9
→ (5, 9)

200

2x − y = 3
x + y = 11

Add both: 3x = 14 → x = 14/3
Then y = 11 − 14/3 = 19/3
→ (14/3, 19/3)

300

3x + 2y = 12
4x − y = 10

Solve → x = 4, y = 0
→ (4, 0)

400

5x − 3y = 7
10x − 6y = 14

The second equation is 2× the first → infinitely many solutions (same line).

500

x − 2y = −1
3x + y = 17

From first: x = 2y − 1
Plug in: 3(2y − 1) + y = 17 → 6y − 3 + y = 17
→ 7y = 20 → y = 20/7
Then x = 2(20/7) − 1 = 40/7 − 1 = 33/7
→ (33/7, 20/7)

Exponents & Radicals

100

2³

8

200

x⁴ · x²

x⁶

300

√50 = √(25×2)

5√2

400

(3x²y)² = 9x⁴y²

9x⁴y²

500

2ˣ = 32

32 = 2⁵ → x = 5

Polynomials

100

(2x + 3) + (x − 5)

3x − 2

200

(x + 6)(x + 1)

x² + 7x + 6

300

x² − 9

Difference of squares → (x − 3)(x + 3)

400

4x² − 20x

Factor out 4x → 4x(x − 5)

500

x³ + 27

Sum of cubes → (x + 3)(x² − 3x + 9)