Which ordered pair makes both inequalities true? y > -3x + 3 y ? 2x – 2 (1,0) (-1,1) (2,2)
The Correct Answer and Explanation is:
We are given two inequalities:
- y>−3x+3y > -3x + 3y>−3x+3
- y≤2x−2y \leq 2x – 2y≤2x−2
We are asked to determine which of the following ordered pairs satisfies both inequalities:
- (1,0)(1, 0)(1,0)
- (−1,1)(-1, 1)(−1,1)
- (2,2)(2, 2)(2,2)
Let’s test each ordered pair one by one.
Test (1, 0):
- For the first inequality, y>−3x+3y > -3x + 3y>−3x+3: 0>−3(1)+3⇒0>−3+3⇒0>0(False)0 > -3(1) + 3 \quad \Rightarrow \quad 0 > -3 + 3 \quad \Rightarrow \quad 0 > 0 \quad (\text{False})0>−3(1)+3⇒0>−3+3⇒0>0(False)
- For the second inequality, y≤2x−2y \leq 2x – 2y≤2x−2: 0≤2(1)−2⇒0≤2−2⇒0≤0(True)0 \leq 2(1) – 2 \quad \Rightarrow \quad 0 \leq 2 – 2 \quad \Rightarrow \quad 0 \leq 0 \quad (\text{True})0≤2(1)−2⇒0≤2−2⇒0≤0(True)
Since the first inequality is false, the pair (1,0)(1, 0)(1,0) does not satisfy both inequalities.
Test (-1, 1):
- For the first inequality, y>−3x+3y > -3x + 3y>−3x+3: 1>−3(−1)+3⇒1>3+3⇒1>6(False)1 > -3(-1) + 3 \quad \Rightarrow \quad 1 > 3 + 3 \quad \Rightarrow \quad 1 > 6 \quad (\text{False})1>−3(−1)+3⇒1>3+3⇒1>6(False)
- For the second inequality, y≤2x−2y \leq 2x – 2y≤2x−2: 1≤2(−1)−2⇒1≤−2−2⇒1≤−4(False)1 \leq 2(-1) – 2 \quad \Rightarrow \quad 1 \leq -2 – 2 \quad \Rightarrow \quad 1 \leq -4 \quad (\text{False})1≤2(−1)−2⇒1≤−2−2⇒1≤−4(False)
Since both inequalities are false, the pair (−1,1)(-1, 1)(−1,1) does not satisfy both inequalities.
Test (2, 2):
- For the first inequality, y>−3x+3y > -3x + 3y>−3x+3: 2>−3(2)+3⇒2>−6+3⇒2>−3(True)2 > -3(2) + 3 \quad \Rightarrow \quad 2 > -6 + 3 \quad \Rightarrow \quad 2 > -3 \quad (\text{True})2>−3(2)+3⇒2>−6+3⇒2>−3(True)
- For the second inequality, y≤2x−2y \leq 2x – 2y≤2x−2: 2≤2(2)−2⇒2≤4−2⇒2≤2(True)2 \leq 2(2) – 2 \quad \Rightarrow \quad 2 \leq 4 – 2 \quad \Rightarrow \quad 2 \leq 2 \quad (\text{True})2≤2(2)−2⇒2≤4−2⇒2≤2(True)
Since both inequalities are true, the pair (2,2)(2, 2)(2,2) satisfies both inequalities.
Conclusion:
The correct ordered pair that satisfies both inequalities is (2,2)(2, 2)(2,2).
