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Equation Solving
LessThan is the twelfth in a series of fifteen puzzle caches of increasing difficulty based on solving equations. It is recommended that one start with Abacus (GC6A006) and continue in alphabetical sequence through Brackets (GC6AR81), Compass (GC6AWD0), Decagon (GC6C4EE), Ellipse (GC6CFW3), FOIL (GC6HDNE), Googol (GC751JK), Hectare (GC75BD3), Index (GC75EAM), Jargon (GC75X8W), Kite (GC76010), LessThan (GC763HV ), Median (GC7DH17), Nonagon (GC7DWC4) and Obelus (GC7DWCD).
The Name - LessThan
The mathematical symbol for "less than” is <. The symbol can be used to say that eight is less than nine: 8 < 9. Note that it is like an equals sign (=) but towards the smaller number the two horizontal lines are so close together they touch - an indication that the adjacent number is smaller than the one at the other end of the symbol. We chose this name for this cache because it starts with “L” and it gives us an excuse to leave the world of “equations” briefly for that of “inequations.” We hope you enjoy the detour!
LessThan - the Cache
Symbols Used in Inequalitites and Inequations
There are several inequality symbols that we need to define:
< means “is less than” as we have noted above;
≤ means “is less than or equal to;"
> means “is greater than;”
≥ means “is greater than or equal to.”
Each of these has a negative equivalent which we will just list because we are not going to concern ourselves with them (and those slashes ought not to be there but I can't eliminate them!):
≮ (not less than);
≰ (not less than nor equal to);
≯ (not greater than);
≱ (not greater than nor equal to).
Some Differences Between Solutions for Equations and Inequations
For these illustrations, we will use the inequation: 12 > 8
Adding to Both Sides
12 > 8
12 + 5 ? 8 + 5
17 > 13 . . . That seems to work fine.
Subtracting from Both Sides (or adding a negative, if you prefer)
12 > 8
12 - 3 ? 8 - 3
9 > 5 . . . That seems alright too.
Multiplying Both Sides by a Positive Number
12 > 8
(3)(12) ? (3)(8)
36 > 24 . . . That works the same way as equations.
Dividing Both Sides by a Positive Number
12 > 8
12/4 ? 8/4
3 > 2 . . . And that seems fine too.
BUT When Multiplying Both Sides by a Negative Number
12 > 8
(-5)(12) ? (-5)(8)
- 60 < - 40 . . . Note that in order to make a true statement, the inequality sign has had to be reversed!
AND When Dividing Both Sides by a Negative Number
12 > 8
12/(-2) ? 8/(-2)
- 6 < - 4 . . . Note that in order to make a true statement, the inequality sign has had to be reversed!
Summary
The process for solving inequations is essentially the same as that for solving equations except that as soon as one indicates multiplication or division by a negative number, the inequality sign has to be reversed
A Couple of Definitions
Integers
In this cache, we are going to deal only with integers. By integers, we mean all the numbers that we use to count things, that is 1, 2, 3, 4, and so on; their negative counterparts, (-1), (-2), (-3), (-4), and so on and 0. We use “I” to stand for “Integers” and write them as follows:
I = {. . . , -3, -2, -1, 0, 1, 2, 3, . . .}
The brace brackets used are translated as “the set of all” and the dots are used to indicate that the numbers go on forever - and we express the dots as "and so on."
Solution Sets
With an equation, one gets one answer for x - at least in our experience so far.
3x = 15
x = 15/3
x = 5
With an inequation, one gets a lot of answers.
3x > 15
x > 15/3
x > 5 . . . meaning that x is anything greater than 5.
We write all the answers using a “solution set” and the brace brackets introduced above. So the answer to this inequation would be written:
The solution set is {6, 7, 8, . . .}.
It would be spoken:
“The solution set is the set of all 6, 7, 8 and so on.”
Some Examples of Inequation Solutions
Example 1
5x + 4 ≥ 19
5x ≥ 19 - 4
5x ≥ 15
x ≥ 15/5
x ≥ 3
So, the solution set is {3, 4, 5, 6, . . . }
(Note that since the inequality sign is ≥, meaning “greater than or equal to,” 3 is included in the solution set.)
Example 2
4x - 7 < 9
4x < 9 + 7
4x < 16
x < 16/4
x < 4
So, the solution set is {3, 2, 1, 0, -1, . . .}
(Note that since the inequality sign is <, meaning “less than,” 4 is not included in the solution set.)
Example 3
7x - 13 ≤ 5x + 5
7x - 5x ≤ 5 + 13
2x ≤ 18
x ≤ 18/2
x ≤ 9
So, the solution set is {9, 8, 7, 6, . . .}
(Again, note that since the inequality sign is ≤, meaning “less than or equal to,” 9 is included in the solution set.)
Example 4
2a + 9 > 24 + 7a
2a - 7a > 24 - 9
- 5a > 15
a < 15/(-5) (Note that we divided both sides by a negative so the inequality sign reverses.)
a < -3
So, the solution set is {- 4, -5, -6, -7, . . . }. (-3 is not included.)
Example 5
2p/3 ≤ 5/12 + 3p/4
Multiply by 12 to eliminate denominators.
(12)(2p)/3 ≤ (12)(5)/12 + (12)(3p)/4
(4)(2p) ≤ 5 + (3)(3p)
8p ≤ 5 + 9p
8p - 9p ≤ 5
- p ≤ 5
-p/(-1) ≥ 5/(-1) (Note the reversal in the inequality sign due to dividing by a negative.)
p ≥ -5
So, the solution set is {-5, -4, -3, -2, -1, 0, 1, . . .}
And Now Your Questions
If necessary, please remind yourselves how to “clear” fractions and decimals by checking GC75BD3 and GC75X8W respectively.
Remember that we are working with the set of integers only - see above.
Question 1
4a - 15 ≥ - 7
The third number in the solution set which we will call A = ___.
Question 2
3b + 6 ≤ 9
The fourth number in the solution set which we will call B = ___.
Question 3
5c - 3 > 7c + 5
The third number in the solution set which we will call C = ___.
Question 4
6d + 11 < 11d + 1
The first number in the solution set which we will call D = ___.
Question 5
5e - 7 + 2e ≤ - 2 + 3e + 3
The largest number in the solution set which we will call E = ___.
Question 6
4f - 4/5 - 8/15 - 7f > 2f + 13/15 - 6f - 1/5
The third number in the solution set which we will call F = ___.
Question 7
0.27 - 0.3g - 0.8 + 0.6g < 0.8g - 0.53 + 0.1 - 0.4g
The smallest number in the solution set which we will call G = ___.
The Location of the Cache
A = ___ , B = ___ , C = ___ , D = ___ , E = ___ , F = ___ , G = ___ .
The co-ordinates of the cache are N 44̊ ab.cde’ and W 078̊ fg.hij’ where none of the lower case letters is related to any of the letters above except as defined below:
a = F - D = ___ ; b = E(A + B) = ___ ; c = - (B)(D) = ___ ; d = G + E = ___ ; e = E(B + D) = ___;
f = (G)(F) + A = ___ ; g = G + B(A + C) = ___ ; h = (-D)(E) - C = ___ ; i = - (A + C) = ___ ; j = E(F + B) = ___ .
You Will Find the Cache at:>
N 44̊ __ __ . __ __ __’ and W 078̊ __ __ . __ __ __’
Additional Comments
- Please bring your own writing utensil;
- Be aware of the possibility of others on the trail;
- Please replace the cache as hidden;
- Please feel free to check your answers for this puzzle on GeoChecker.com.