Solving Inequalities Activity Fall {Autumn Algebra 1 and 2 Activity Worksheet} - Free Printable

A. 5x + 2 < x - 10
Subtract x from both sides: 4x + 2 < -10
Subtract 2 from both sides: 4x < -12
Divide by 4: x < -3

B. -2(x - 1) > 6
Distribute: -2x + 2 > 6
Subtract 2: -2x > 4
Divide by -2 (reverse inequality): x < -2

C. 3(x + 3) ≤ 7x + 2
Distribute: 3x + 9 ≤ 7x + 2
Subtract 3x: 9 ≤ 4x + 2
Subtract 2: 7 ≤ 4x
Divide by 4: 7/4 ≤ x or x ≥ 1.75

D. 2(3x - 1) ≤ 4(x - 2)
Distribute: 6x - 2 ≤ 4x - 8
Subtract 4x: 2x - 2 ≤ -8
Add 2: 2x ≤ -6
Divide by 2: x ≤ -3

E. 3x + 6 ≥ -12
Subtract 6: 3x ≥ -18
Divide by 3: x ≥ -6

F. 17 > 3 - 2(5 + 2)
Simplify inside parentheses: 17 > 3 - 2(7)
Multiply: 17 > 3 - 14
Add: 17 > -11
This is always true, so all real numbers satisfy this inequality.

G. 8x - 2 ≤ 5x + 7 - 2x + 1
Combine like terms on right: 8x - 2 ≤ 3x + 8
Subtract 3x: 5x - 2 ≤ 8
Add 2: 5x ≤ 10
Divide by 5: x ≤ 2

H. x/3 < 2(2x + 9) + 12
Distribute: x/3 < 4x + 18 + 12
x/3 < 4x + 30
Multiply both sides by 3: x < 12x + 90
Subtract 12x: -11x < 90
Divide by -11 (reverse inequality): x > -90/11 or x > -8.1818...

I. 5 - (x - 7) ≥ 5x + x
Simplify: 5 - x + 7 ≥ 6x
12 - x ≥ 6x
Add x: 12 ≥ 7x
Divide by 7: 12/7 ≥ x or x ≤ 12/7

J. 3(x + 4) - 2(2x + 5) ≤ 7
Distribute: 3x + 12 - 4x - 10 ≤ 7
Combine: -x + 2 ≤ 7
Subtract 2: -x ≤ 5
Multiply by -1 (reverse inequality): x ≥ -5

Now, match the inequalities to the letters:
A: x < -3 → x < -3
B: x < -2 → x < -2
C: x ≥ 1.75 → x ≥ 1.75
D: x ≤ -3 → x ≤ -3
E: x ≥ -6 → x ≥ -6
F: always true → all real numbers
G: x ≤ 2 → x ≤ 2
H: x > -90/11 → x > -90/11
I: x ≤ 12/7 → x ≤ 12/7
J: x ≥ -5 → x ≥ -5

The answer is: A. x < -3, B. x < -2, C. x ≥ 1.75, D. x ≤ -3, E. x ≥ -6, F. all real numbers, G. x ≤ 2, H. x > -90/11, I. x ≤ 12/7, J. x ≥ -5