Writing Equations
See Q 1
1. Solution:
(3(r - 4) / 5) - 6 = 12
Add 6 to both sides: 3(r - 4) / 5 = 18
Multiply both sides by 5: 3(r - 4) = 90
Divide by 3: r - 4 = 30
Add 4: r = 34
See Q 4
4. Solution:
a) Equation: 14.99x + 30.00 = 19.99x + 15.00
Subtract 14.99x: 30.00 = 5.00x + 15.00
Subtract 15.00: 15.00 = 5.00x
Divide by 5.00: x = 3
b) It will take 3 months for both streaming service plans to cost the same price.
See Q 6
6. Solution:
Q = m c T
Divide both sides by (mc): T = Q / (m c)
See Q 9
Solution:
-2(x + 4) + 5 ≤ 11
Distribute: -2x - 8 + 5 ≤ 11
Combine terms: -2x - 3 ≤ 11
Add 3: -2x ≤ 14
Divide by -2 (flip inequality): x ≥ -7
See Q2
2. Solution:
3(4 + 3x) = 9x + 15
Distribute: 12 + 9x = 9x + 15
Subtract 9x from both sides: 12 = 15 (False statement)
No Solution
See Q 5
. Solution:
a) Profit = Revenue - Total Cost > 0 ⇒ Revenue > Total Cost
Inequality: 45h > 22.50h + 450
b) Subtract 22.50h from both sides: 22.50h > 450
Divide by 22.50: h > 20
The minimum number of hoodies the club must sell to make a profit is 21 hoodies.
See Q 7
7. Solution:
d = v t + (1/2) a t²
Subtract vt: d - v t = (1/2) a t²
Multiply by 2: 2(d - v t) = a t²
Divide by t²: a = 2(d - v t) / t²
See Q 10
. Solution:
-15 < 3x - 3 ≤ 9
Add 3 to all parts: -12 < 3x ≤ 12
Divide all parts by 3: -4 < x ≤ 4
Interval Notation: (-4, 4]
See Q 3
Solution:
(2/3)x - 3/4 = (1/2)x
Clear fractions by multiplying all terms by LCD = 12:
12 * (2/3)x - 12 * (3/4) = 12 * (1/2)x
8x - 9 = 6x
Subtract 8x: -9 = -2x
Divide by -2: x = 9/2 (or 4.5)
See Q 12
12. Solution:
V/I= R = 4
See Q 8
8. Solve for b:
b=d/a
See Q 11
. Solution:
Left inequality: 4x + 2 > 22 ⇒ 4x > 20 ⇒ x > 5
Right inequality: 3(2x - 1) < 9 ⇒ 6x - 3 < 9 ⇒ 6x < 12 ⇒ x < 2
Solution: x < 2 OR x > 5
Graph: Open circle at 2 pointing left, open circle at 5 pointing right.