Set Builder Notation
−8p − 16 < −5p + 7p
{p ∣ p > −8/5}
Maria earns $15 per hour babysitting. She wants to buy a pair of headphones that cost $120. Write and solve an inequality that shows how many hours she must work to afford them.
h ≥ 8
4x + 6 > 10
x > 1
6 - 2x > 20
x < -7
-2x + 5 > 7 or -4x + 6 < 3
x < -1 or x > 3/4
3(2m − 1) > 4m − 7
{m ∣ m > −2}
You have $42 to spend on food for your dog. Each bag costs the same amount. If you want to buy 6 bags, write and solve an inequality that represents the price per bag.
p ≤ 7
5x − 7 ≤ 2x + 8
x ≤ 5
-2x − 4 ≤ -8x
x ≤ 2/3
10 < x(2 + 5) < -2
10/7 < x < -2/7
6x + 9 ≤ 3(2x + 4) + 5
All Real solutions
A family is driving to visit relatives who live 300 miles away. If the car travels at a speed of no more than 60 miles per hour, write and solve an inequality that represents the travel time.
t ≥ 5
−3.5 < 0.5(4x+17) < 2.5
−6 < x < −3
1/3(6x − 3) − 5 ≤ 1/3(6 − 9x)
x ≤ 8/5
35 < 7(2 − b) or 1/3(15b − 12) ≥ 21
b <−3 or b ≥ 5
2(y + 4) + 5 > 4y − 3
{ y | y < 8}
Alex is buying a birthday gift for each of his three cousins. He also has to pay a flat $5 shipping fee. If he has a budget of $100, write and solve an inequality to find how much he can spend on each gift.
g ≤95 / 3
3(1 − 2x) ≥ 3 − 6x
Infinite Solutions
-4x + 2x + 3 > 6x - 4
x < 7/8
5x + 1 < x + 3 < 12
x < 1/2
7x − 12 ≤ 2x + 8
{ x | x ≤ 4 }
A nutritionist recommends eating between 1800 and 2200 calories per day. If someone has already eaten 1350 calories, write and solve an inequality to show how many calories they can eat for dinner.
450 ≤ d ≤ 850
1/2(4x − 2) − 2/3(6x + 9) ≤ 4
x ≥ - 11/2
7(1/2k + 6.5) − 2k ≤ 3(k + 11/3) + 37.5
k ≥ −2
-5 < 4x + 2(3x - 2) < 10
-1/10 < x < 7/5