Question
y ≤3x +1
x – y >1
Which of the following ordered pairs ( x, y)satisfies the system of inequalities above?
- (−2, −1 )
- (−1, 3)
- (1, 5)
- (2, −1)
Hint:
Hint:- Inequalities are the mathematical expressions in which both sides are not
equal. In inequality, unlike in equations, we compare two values. The equal sign
in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to),
or not equal to sign.
There are two methods by which we can solve this question.
The correct answer is: (2, −1)
Solution:- Option D is correct.
Method 1:-
We have given two inequalities
y ≤3x +1
x – y >1
Any point (x, y) that is a solution to the given system of inequalities must
satisfy both inequalities in the system. Since the second inequality in the system can be rewritten as y < x − 1, the system is equivalent to the following system.
y ≤3x +1
x – y >1
Since , 3x + 1 > x − 1 for x > −1
and 3x + 1 ≤ x − 1 for x ≤ −1,
it follows that y < x − 1 for x > −1 and y ≤ 3x + 1 for x ≤ −1. Of the given choices,
only (2, −1) satisfies these conditions because −1 < 2 − 1 = 1
Method 2:-
Substituting (2, −1) into the first inequality gives
−1 ≤ 3(2) + 1, or −1 ≤ 7,
which is a true statement.
Substituting (2, −1) into the second inequality gives
2 − (−1) > 1, or 3 > 1,
which is a true statement.
Therefore, since (2, −1) satisfies both inequalities,
it is a solution to the system.
- If we considered Option A , it is incorrect because substituting −2 for x and −1
for y in the first inequality gives −1 ≤ 3(−2) + 1, or −1 ≤ −5, which is false.
- If we considered option B , it is incorrect because substituting −1 for x and 3 for
y in the first inequality gives 3 ≤ 3(−1) + 1, or 3 ≤ −2, which is false.
- If we considered option C, it is incorrect because substituting 1 for x and 5 for
y in the first inequality gives 5 ≤ 3(1) + 1, or 5 ≤ 4, which is false.
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As opposed to this, a circle's area indicates the space it occupies.
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Therefore, to calculate a circle's circumference, we must know its radius or diameter.
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where,
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π is a mathematical constant with an estimated value of 3.14 (to the nearest two decimal places).
