dave13 wrote:
14101992 wrote:
The number of n satisfying -n+2≥0 and 2n≤5 is
A. 0
B. 1
C. 2
D. 3
E. None of the above
Greetings
generis,
could you change in this question some numbers "The number of n satisfying -n+2≥0 and 2n≤5 " in which answer would 2, or 3 or any number and explain solution so as I compare with the question to which answer E
please
i just wanna cement inequility in my mind
cause i feel a bit of running sands in inequlity
have an terrific day
Hi
dave13 ,
I wonder whether the real issue here is a concept that gets muddied by the prompt.
The prompt says: The number of n satisfying -n+2≥0 and 2n≤5 is[?]
Let me try a rewritten prompt; if it does not help, no problem.
We could call the numbers we are looking for
n. We could call them
n and
N. We could call them
x and
y.
The variables, no matter what they are, mean "integers, decimals, and fractions."
There are two inequalities.
Only some numbers, denoted by \(x\) and \(y\) (including integers, decimals, and fractions, but not imaginary numbers) will follow the rule each inequality "defines."
How many numbers \(x\) and \(y\) follow the rule set out by the inequality -x+2≥0, and follow the rule set out by the inequality 2y≤5?
First inequality's rule\(-x+2≥0\)We need x by itself. Subtract 2 from both sides:
\(-x ≥ -2\), which equals
\((-1)*(x) ≥ (-1)*(2)\)We need x to be positive. Divide both sides by (-1).
When we divide (or multiply) an inequality by a negative number, we must flip the direction of the sign. (Long story. Trust me?) Thus:
\(\frac{(-1)*(x)}{(-1)}\)≤ \(\frac{(-1)(2)}{(-1)}\) , which equals
\(x ≤ 2\)How many integers, decimals, and fractions are RULE -> less than or equal to 2?
Here are just a few samples. Every example follows the rule:
2 is equal to 2
These numbers are all less than 2, so they are okay, they follow the rule: 0, -\(\frac{3}{4}\), - 10.7, -2,801 . . . I can't finish this list. It is endless.
One part of the answer to, "How many numbers, \(x\), satisfy the inequality
\(x ≤ 2\)?"
An infinite number of numbers.
Second inequality's ruleHow many numbers, \(y\), follow the rule set out by
\(2y ≤ 5\)No pesky negative values to worry about. Just get \(y\) by itself. Divide both sides by 2
\(\frac{2y}{2} ≤ \frac{5}{2}\)\(y ≤ 2.5\)How many numbers, \(y,\) follow the rule that the number, \(y\), is less than or equal to 2.5?
2.5 equals 2.5
2.499 is less than 2.5
2.4 is less than 2.5
2\(\frac{1}{4}\) is less than 2.5
2.156792.00001 is less than 2.5
2 is less than 2.5
These numbers also all follow the rule for
\(y ≤ 2.5\):
0, -
\(\frac{3}{4}\), -10.79, -2,801 . . . endless
Compare number lines. Blue = "allowed, follows the rule," Red = "now allowed, breaks the rule"
\(x ≤ 2\)<<-------0-----1------2-----2.5----3--->>\(y ≤ 2.5\)<<-------0-----1------2-----2.5-----3--->>We have a problem. ALL of the numbers that are "okay" for the first inequality are okay for the second inequality.
Every number that falls on the blue parts is fine, follows both inequality rules.
BUT the reverse is not true. Some numbers that satisfy the second inequality DO NOT satisfy the first inequality.
Between 2 and 2.5, one line says RED (no) and one line says BLUE (yes)
The numbers in green, above? They do NOT follow
\(x ≤ 2\)Because of THIS more restrictive inequality,
\(x ≤ 2\),
the numbers that
were green above now have to be red:
2.00001, 2.15679, 2\(\frac{1}{4}\), 2.4, 2.499, 2.5Those numbers satisfy
\(y ≤ 2.5\) but not
\(x ≤ 2\):
So we have to follow the stricter rule. In a way, that means
\(y ≤ 2.5\) is irrelevant.
Those numbers that
were green and are
now red are all to the right of 2, greater than 2. Not allowed.
So we "chop off" part of the number line that we thought would answer the question properly. That part does not. It violates
\(x ≤ 2\).
<<______0________________2____2.5However, what range of answers still works? This one:
<<_-2,801___-10.7__\(-\frac{3}{4}\)__0____2How many numbers are less than or equal to 2? An infinite number of numbers.
No answer choices says "infinite."
The answer is E.
I hope that helps.
Sometimes, when I feel as if I'm in quicksand while reviewing a topic, I go away from it for a couple of days.
I admire your tenacity. I hope this helps.
_________________
People must live and create. Live to the point of tears.
—Albert Camus