# SOLVING INEQUALITIES PRACTICE FOR SAT

Problem 1 :

An online office supply store sells pens at p dollars per box and offers free shipping on orders of \$75 or more. If John decides to order new pens from this store, which of the following inequalities expresses the number of boxes n that he must purchase for the order to qualify for free shipping ?

a)  n ≤ (p/75)        b)  n  ≥ p/75    c) n ≤ 75/p    d)  n ≥ 75/p

Solution :

Cost per box = p

number of boxes he orders = n

Cost of his purchase = np

np ≥ 75

Dividing by p on both side, we get

≥ 75/p

Problem 2 :

≤ 3x + 1

x - y > 1

Which of the following ordered pairs (x, y) satisfies the system of inequalities above ?

a)  (-2, -1)    b) (-1, 3)    c) (1, 5)    d)  (2, -1)

Solution :

≤ 3x + 1

x - y > 1

Applying option a :

 y ≤ 3x + 1x = -2 and y = -1-1 ≤ 3(-2) + 1-1 ≤ -6 + 1-1 ≤ -5False x - y > 1x = -2 and y = -1-2 - (-1) > 1-2 + 1 > 1-1 > 1False

So, option a is not correct.

Applying option b :

 y ≤ 3x + 1x = -1 and y = 33 ≤ 3(-1) + 13 ≤ -3 + 13 ≤ -2False x - y > 1x = -1 and y = 33 - (-1) > 13 + 1 > 14 > 1True

So, option b is not correct.

Applying option c :

 y ≤ 3x + 1x = 1 and y = 55 ≤ 3(1) + 15 ≤ 3 + 15 ≤ 4False x - y > 1x = -1 and y = 33 - (-1) > 13 + 1 > 14 > 1True

Applying option d :

 y ≤ 3x + 1x = 2 and y = -1-1 ≤ 3(2) + 1-1 ≤ 6 + 1-1 ≤ 7True x - y > 1x = 2 and y = -12 - (-1) > 13 > 1True

So, option d is correct.

Problem 3 :

Which of the following numbers is NOT a solution of the inequality 3x − 5 ≥ 4x −3 ?

a) −1     b) −2     c) −3      d) −5

Solution :

3x − 5 ≥ 4x −3

Subtracting 4x on both sides,

3x - 4x - 5 ≥ −3

-x ≥ −3 + 5

-x ≥ 2

Dividing by - on both sides.

≤ -2

(-, -2] are solutions. In the solution set -1 is not on one of the values in solution set. So, option a is not a solution.

Problem 4 :

y < -x + a, y > x + b

In the xy plane if (0, 0) is a solution to the system of inequalities above, which of the following relationships between a and b must be true ?

a)  a > b    b) b > a     c) |a| > |b|      d) a = -b

Solution :

Since (0, 0) is a solution for the inequalities given,

y < -x + a, y > x + b

0 < -0 + a and 0 > 0 + b

0 < a and 0 > b

Then a > b, option a is correct.

Problem 5 :

≤ -15x + 3000

y ≤ 5x

In the xy plane, if the point with coordinates (a, b) lies in the solution to the set of inequalities above, what is the maximum possible value of b ?

Solution :

≤ -15x + 3000 ----(1)

y ≤ 5x ----(2)

Converting the inequalities into equation, we get

y = -15x + 3000

y = 5x

-15x + 3000 = 5x

-15x - 5x = -3000

-20x = -3000

x = 3000/20

x = 150

Applying the value of x in (1)

≤ -15(150) + 3000

≤ -2250 + 3000

≤ 750

So, the maximum value of b is 750.

Problem 6 :

The total cost, in dollars, to rent a surfboard consists of a \$25 service fee and a \$10 per hour rental fee. A person rents a surfboard for t hours and intends to spend a maximum of \$75 to rent the surfboard. Which inequality represents this situation?

A) 10t ≤ 75     B) 10 + 25t ≤ 75    C) 25t ≤ 75

D) 25 + 10t ≤ 75

Solution :

Total cost = service fee + cost per hour (number of hours)

Service fee = \$25

Charge per hour = \$10

Let t be the number of hours.

25 + 10t

The maximum rent is \$75

25 + 10t ≤ 75

Problem 7 :

A psychologist set up an experiment to study the tendency of a person to select the first item when presented with a series of items. In the experiment, 300 people were presented with a set of five pictures arranged in random order. Each person was asked to choose the most appealing picture. Of the first 150 participants, 36 chose the first picture in the set. Amongthe remaining 150 participants, p people chose the first picture in the set. If more than 20% of all participants chose the first picture in the set, which of the following inequalities best describes the possible values of p ?

A) p > 0.20 (300−36),where p ≤ 150

B) p > 0.20 (300+36), where p ≤ 150

C) p - 36 > 0.20 (300), where p ≤ 150

D) p + 36 > 0.20 (300), where p ≤ 150

Solution :

Given that among the first 150 participants, 36 chose the first picture in the set. Of the remaining 150, p people chose the first picture in the set (p≤150).

Hence, the total number of people who chose the first picture in the set is (p + 36), p≤150.

If the number of participants who chose the first picture is more than the total number of participants i.e. 300, then, it is represented by the following inequality

(p+36) ≥ 0.20(300), where p ≤ 150

So, option d is correct.

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