Video: Optimiziation Using Inequalities

Given that −1 ≤ (−6𝑥/10) − 1 ≤ 5, find the greatest possible value of 𝑥 + 8.

03:21

Video Transcript

Given that negative one is less than or equal to negative six 𝑥 over 10 minus one which is less than or equal to five, find the greatest possible value of 𝑥 plus eight.

Two things have to happen here. First, we need to find the range of 𝑥 and choose its greatest possible value. And whatever 𝑥-value we find, we need to plug in to the expression 𝑥 plus eight. We’ll start by solving our inequality.

Because we have a fraction in our inequality, it’s easier if we get rid of that first. To move that 10 out of the denominator, I’m going to multiply the entire equation by 10. 10 times negative one equals negative 10. Bring down the less than or equal to. Multiplying 10 by negative six 𝑥 over 10. The 10s cancel out, leaving us with negative six 𝑥. 10 times negative one equals negative 10. 10 times five equals 50. Bring down the less than or equal to sign.

That step will make solving for 𝑥 a simpler process. We’re trying to isolate 𝑥. We need to get rid of this minus 10 by adding 10. But to keep our inequality balanced, we’ll have to do that on the left- and right-hand side as well. Negative 10 plus 10 cancels out. The middle of the inequality now only has negative six 𝑥. On the left, negative 10 plus 10 equals zero. Bring down the sign. On the right, 50 plus 10 equals 60. Bring down the sign. 𝑥 is being multiplied by negative six.

To isolate that 𝑥, we’ll need to divide it by negative six. And that means we’ll have to divide by negative six on the left- and right-hand sides as well. When we multiply or divide with negatives and we’re working with inequalities, we have to flip the sign. In the middle, negative six divided by negative six equals one. There’s only 𝑥 left. On the left, zero divided by negative six equals zero. 60 divided by negative six equals negative 10.

Our inequality says zero is greater than or equal to 𝑥, which is greater than or equal to negative 10. Let’s plug this on a number line: zero and negative 10. 𝑥 can be equal to negative 10. We know that 𝑥 is greater than or equal to negative 10. But 𝑥 is less than or equal to zero. So this is the graph of all the things 𝑥 can be.

What is the greatest value 𝑥 could be? Zero. Every other value for 𝑥 is negative. Now, we can move on to checking the expression 𝑥 plus eight. If 𝑥’s largest possible value is zero, then the largest value of 𝑥 plus eight is eight.

Given these limits for 𝑥, the greatest possible value of 𝑥 plus eight equals eight.

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