Inequalities

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These are the solutions to the CSEC past questions on the topics: Inequalities.
When applicable, the TI-84 Plus CE calculator (also applicable to TI-84 Plus calculator) solutions are provided for some questions.
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Notes and Rules

(1.) At least 5 means \ge 5
It means that the minimum should be 5

(2.) At most 5 means \le 5
It means that the maximum should be 5

(3.) Just 5 means = 5
It means exactly 5 or equal to 5

(4.) More than 5 means \gt 5

(5.) Less than 5 means \lt 5

(6.) No more than 5 means \le 5
It should not be more than 5
It means 5 or less

(7.) No less than 5 means \ge 5
It should not be less than 5
It means 5 or more

Rules of Inequalities
c, d, e are real numbers

$ (1.)\:\:If\:\: c \lt d \:\:and\:\: d \lt e, \:\:then\:\: c \lt e ...Transitive\:\:Rule \\[3ex] (2.)\:\:If\:\: c \gt d \:\:and\:\: d \gt e, \:\:then\:\: c \gt e ...Transitive\:\:Rule \\[3ex] (3.)\:\:If\:\: c \lt d, \:\:then\:\: d \gt c \\[3ex] (4.)\:\:If\:\: c \gt d, \:\:then\:\: d \lt c \\[3ex] (5.)\:\:If\:\: c \lt d, \:\:then\:\: -c \gt -d \\[3ex] (6.)\:\:If\:\: c \gt d, \:\:then\:\: -c \lt -d \\[3ex] (7.)\:\:If\:\: c \lt d, \:\:then\:\: \dfrac{1}{c} \gt \dfrac{1}{d} \\[5ex] (8.)\:\:If\:\: c \gt d, \:\:then\:\: \dfrac{1}{c} \lt \dfrac{1}{d} \\[5ex] (9.)\:\:If\:\: c \lt d, \:\:then\:\: (c + e) \lt (d + e) \\[3ex] (10.)\:\:If\:\: c \gt d, \:\:then\:\: (c + e) \gt (d + e) \\[3ex] (11.)\:\:If\:\: c \lt d, \:\:then\:\: (c - e) \lt (d - e) \\[3ex] (12.)\:\:If\:\: c \gt d, \:\:then\:\: (c - e) \gt (d - e) \\[3ex] (13.)\:\:If\:\: c \lt d, \:\:and\:\: e \gt 0; \:\:then\:\: ce \lt de \\[3ex] (14.)\:\:If\:\: c \lt d, \:\:and\:\: e \lt 0; \:\:then\:\: ce \gt de \\[3ex] (15.)\:\:If\:\: c \gt d, \:\:and\:\: e \gt 0; \:\:then\:\: ce \gt de \\[3ex] (16.)\:\:If\:\: c \gt d, \:\:and\:\: e \lt 0; \:\:then\:\: ce \lt de \\[3ex] (17.)\:\:If\:\: c \lt d, \:\:and\:\: e \gt 0; \:\:then\:\: \dfrac{c}{e} \lt \dfrac{d}{e} \\[5ex] (18.)\:\:If\:\: c \gt d, \:\:and\:\: e \gt 0; \:\:then\:\: \dfrac{c}{e} \gt \dfrac{d}{e} \\[5ex] (19.)\:\:If\:\: c \lt d, \:\:and\:\: e \lt 0; \:\:then\:\: \dfrac{c}{e} \gt \dfrac{d}{e} \\[5ex] (20.)\:\:If\:\: c \gt d, \:\:and\:\: e \lt 0; \:\:then\:\: \dfrac{c}{e} \lt \dfrac{d}{e} $

Formula Sheet: List of Formulae

(2.) Lisa has $56 to buy a total of no more than 70 red balloons and green balloons for her party.
She buys more green balloons than red red balloons but must buy at least 15 red balloons.
Each red balloon costs $0.75 and each green green balloon costs $0.50
Let x and y represent the number of red balloons and the number of green balloons respectively.
Write TWO inequalities in x and y, other than x ≥ 0 and y ≥ 0, to represent the information above.

1st: Lisa wants to buy a total of no more than 70 red balloons and green balloons for her party.
No more than 70 means ≤ 70

$ x + y \le 70...\text{1st Inequality} \\[3ex] $ 2nd: Lisa has $56 to buy ... red balloons and green balloons ...
Each red balloon costs $0.75 and each green green balloon costs $0.50
The costs of the balloons (red and green balloons) will not exceed $56
The costs will be up to $56

$ 0.75x + 0.5y \le 56 ...\text{2nd Inequality} \\[3ex] $ These are two inequalities in x and y

But, if you want more, these two inequalities will also work for the question.

3rd: She buys more green balloons than red red balloons ...

$ y \gt x ...\text{3rd Inequality} \\[3ex] $ 4th: She buys more green balloons than red red balloons but must buy at least 15 red balloons.
Each red balloon costs $0.75 and each green green balloon costs $0.50
At least means ≥
Assume 15 red balloons (must buy at least 15 red ballons) and 16 green balloons (more green balloons than red balloons, we do not have a half-balloon; balloons come in whole numbers)

Cost of 15 red balloons = 15(0.75) = $11.25
Cost of 16 green balloons = 16(0.5) = $8
Cost of 15 red balloons and 16 green balloons = $11.25 + $8 = $19.25
This is the minimum amount she must spend.

$ 0.75x + 0.5y \ge 19.25 ...\text{4th Inequality} $

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