Video: Identifying the Point That Is Not a Solution to a Given Inequality

Which of the following points is not a solution of the inequality −3𝑥 − 3𝑦 ≤ 5? [A] (8, 0) [B] (−1, −4) [C] (6, 3) [D] (6, 0) [E] (−5, 8)

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Video Transcript

Which of the following points is not a solution of the inequality negative three 𝑥 minus three 𝑦 is less than or equal to five? And the options are (A) eight, zero; (B) negative one, negative four; (C) six, three; (D) six, zero; and (E) negative five, eight.

So if we have a pair of coordinates, the first coordinate represents our 𝑥-value and the second coordinate represents our 𝑦-value. So what we can do is we can substitute our 𝑥- and 𝑦-values from each of our answers into our inequality to see if it satisfies what we’ve got. So I’m gonna start with eight, zero. So as we’ve already said, our 𝑥-coordinate or 𝑥-value is gonna be eight and our 𝑦-value is gonna be zero. So if we substitute these in, we’re gonna get negative three multiplied by eight minus three multiplied by zero, because this is the left-hand side of our inequality. Which is gonna give us negative 24 minus zero, which is just gonna give us negative 24.

Well, if we check to see if it satisfies our inequality, so if it’s the solution of the inequality. We’d need negative 24 to be less than or equal to five. Well, negative 24 is less than or equal to five. So, therefore, we can say that (A) is a solution of our inequality. So, therefore, it’s not the correct answer because what we’re looking for is one of the following points which is not a solution. Okay, so now let’s move on to answer (B).

Well, with (B), we’ve got negative three multiplied by negative one minus three multiplied by negative four, again, cause we have our 𝑥- and 𝑦-values, respectively. Now, as you could see here, I’ve actually put parentheses around these. And the reason I’ve done this is because if we’re typing it into our calculator, for instance, if we do negative three multiplied by negative one, if we don’t put a parenthesis in in front of the negative one, it will actually sometimes give us the wrong answer. So it’s just something that might be useful if you are using a calculator.

So what we’re gonna get is three minus negative 12. And the reason we get that is because if we have a negative multiplied by a negative, then this is equal to a positive. So negative three multiplied by negative one gives us positive three. Similarly, if we subtract a negative, then it’s the same as adding a positive. So three minus negative 12 is the same as three add 12. So this gives us a result of 15.

Now if we check to see if this satisfies our inequality, well 15 is not less than or equal to five. So, therefore, it does not satisfy our inequality. Well, if we remind ourselves what the problem’s looking for. Well, it’s looking for a following points which are not a solution of the inequality. Well, therefore, this could be one of our correct answers, (B), because it is not a solution of our inequality. So let’s check the others anyway to see if they are or are not solutions to the inequality as well.

So what we’re gonna have for (C) is negative three multiplied by six minus three multiplied by three. Which is gonna give us negative 18 minus nine, which is equal to negative 27. Well, if we check to see if this satisfies our inequality, well negative 27 is less than or equal to five. So, therefore, it does satisfy our inequality. Which means answer (C) is not the correct answer because as, again, we remind ourselves that we’re looking for the points that are not a solution of the inequality. So now we can move on to (D).

So what we’re gonna have is negative three multiplied by six minus three multiplied by zero. Which is gonna give us negative 18 minus zero, which is just equal to negative 18. Once again, if we check this, we can see that it does satisfy our inequality because negative 18 is less than or equal to five. So, therefore, answer (D) is not the correct answer to the problem because, again, it is a solution of the inequality. So it is looking like answer (B) is the correct one.

We’re gonna do the final one now, which is answer (E). So what we’re gonna have is negative three multiplied by negative five minus three multiplied by eight. And, once again, we’ve got a negative multiplied by a negative cause we’ve got negative three multiplied by negative five. So this is gonna give us a positive, which is 15, then minus 24. And then this is equal to negative nine. So, once again, if we check this, well negative nine is less than or equal to five. Which means that answer (E) is also not the correct answer because it is a solution to the inequality. Negative three 𝑥 minus three 𝑦 is less than or equal to five.

So, therefore, we can say that the correct answer, so the answer which is not a solution of the inequality negative three 𝑥 minus three 𝑦 is less than or equal to five, is answer (B). Which has the 𝑥- and 𝑦-coordinates negative one, negative four, respectively.

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