Inequalities
Give the interval notation and graph of: 7 < x + 4
(∞ , 3)
x > 3 (unshaded; to the right)
The sum of three consecutive integers is 306. What is the largest integer?
x + (x + 1) + (x + 2) = 306
x = 101
103
Xiello and Xander leave the same house at the same time and travel in opposite directions. Xiello travels at 70 km/h, and Xander at 50 km/h. How long will it take before they are 240 km apart?
70t + 50t = 240
2 hours
A candy store mixes 10 grams of strawberry candy costing Php60 per gram with 5 grams of chocolate candy costing Php150 per gram. What is the price per gram of the mixture?
600 + 750 = 1350
Php 90 per gram
A teacher can grade a stack of papers in 4 hours, while her assistant can do it in 12 hours. If they work together, how long will it take?
x/4 + x/12 = 1 (LCD: 12)
3 hours
Give the 4 properties of inequality
Trichotomy Property, Transitive PI, Addition PI, Multiplication PI
Jas is 16 years younger than Eubie. If the sum of their ages is 30, how old is Jas?
x + (x + 6) = 30
Jas is 7 years old
Jaime rides his bike at 50 km/h. His sister Kate leaves 12 minutes later and rides at 80 km/h in the same direction. After how many minutes will Kate catch Jaime?
50(t + 1/5) = 80t
20 minutes
A scientist mixes a 25% acid solution with a 60-liter 55% acid solution to produce a 40% solution. How many liters of 25% solution are needed?
0.25x + 33 = 0.4x + 24 [25x + 3300 = 40x + 2400]
60 liters
Hans can clean a classroom alone in 3 hours, while his classmate Ciel can finish the same job in 6 hours. If they work together, how long will it take them to finish the job?
x/3 + x/6 = 1 (LCD: 6)
2 hours
Give the interval notation and graph of: x + 1 < −2 or 2x > 8
(−∞,−3) ∪ (4,∞)
Hollow circle at −3, shaded to the left; Hollow circle at 4, shaded to the right
If one number is three times as large as another number and the smaller number is increased by 19, the result is 6 less than twice the larger number. What is the larger number?
x + 19 = 2(3x)-6
5 = x
15
Two runners start at opposite ends of a 90 km trail. Dakila runs at 20 km/h, Sam at 25 km/h. How long will it take them to meet?
20t + 25t = 90
2 hours
A snack shop owner mixes macadamia nuts costing Php150 per kg with almonds costing Php90 per kg. How many kilograms of each should be mixed to form 10 kilograms of mixture worth Php120 per kg?
150x + 900 − 90x = 1200
5 kg macadamia, 5 kg almonds
Sean can complete a mural in 10 hours. He works alone for 2 hours, then his friend Ace helps finish it in 4 more hours total work time. How long does Ace take alone?
6/10 + 4/x = 1 (LCD: 10x)
10 hours
What does the properties of absolute value state for
1. For equalities: If |x| = a
2. Inequalities Involving < and ≤: If |x| < a
3. Inequalities Involving > and ≥: If |x| > a
1. or
2. and
3. or
Mavy is Ryley's father. Mavy is 35 years old. Three years ago, Mavy was four times as old as his son was then. How old is Ryley now?
32 = 4(x - 3)
Ryley is now 11
David takes 3 hours to reach a warehouse. On the return trip, due to traffic, he travels 15 km/h slower and takes 4 hours. How far is the warehouse?
3r = 4(r − 15)
180 km
A laboratory has 80 liters of a 40% salt solution. How many liters of water must be added to make it a 20% solution?
32 = 16 + 0.2x [3200 = 1600 + 20x]
80 liters of water
Clyde can complete a mural in 12 hours. He works with Mathew for 3 hours, then Mathew leaves and Clyde finishes the remaining work alone in 5 hours. How long will it take Mathew to complete the mural alone?
8/12 + 3/x = 1 (LCD: 12x)
9 hours
Solve: |2x + 4| − 6 ≤ 10
−10 ≤ x ≤ 6
The width of a rectangle is 11 cm. Find all possible values for the length of the rectangle if the perimeter is at least 300 cm.
300 ≤ 2(11) + 2(x)
139 ≤ x; At least 139 cm
Aidan travels 6 hours total to a province covering 370 km. It travels at 50 km/h first, then 80 km/h for the rest of the trip. How many hours did he travel at 80 km/h?
50t + 80(6 − t) = 370
3 hours and 40 minutes at 80 km/h
A chemist has 50 grams of a 30% salt solution. How many grams of pure salt must be added to make it a 50% solution?
x + 15 = 25 + 0.5x [100x + 1500 = 2500 + 50x]
20 grams of salt
A tank is filled by one pipe in 4 hours and drained by another in 8 hours. If both are open, how long will it take to fill the tank?
x/4 − x/8 = 1 (LCD: 8)
8 hours