# Question Video: Optimiziation Using Inequalities Mathematics

Given that β1 β€ (β6π₯/10) β 1 β€ 5, find the greatest possible value of π₯ + 8.

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### Video Transcript

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

Two things have to happen here. First, we need to find the range of π₯ and choose its greatest possible value. And whatever π₯-value we find, we need to plug in to the expression π₯ plus eight. Weβll start by solving our inequality.

Because we have a fraction in our inequality, itβs easier if we get rid of that first. To move that 10 out of the denominator, Iβ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 π₯ over 10. The 10s cancel out, leaving us with negative six π₯. 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 π₯ a simpler process. Weβre trying to isolate π₯. We need to get rid of this minus 10 by adding 10. But to keep our inequality balanced, weβ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 π₯. 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. π₯ is being multiplied by negative six.

To isolate that π₯, weβll need to divide it by negative six. And that means weβll have to divide by negative six on the left- and right-hand sides as well. When we multiply or divide with negatives and weβre working with inequalities, we have to flip the sign. In the middle, negative six divided by negative six equals one. Thereβs only π₯ 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 π₯, which is greater than or equal to negative 10. Letβs plug this on a number line: zero and negative 10. π₯ can be equal to negative 10. We know that π₯ is greater than or equal to negative 10. But π₯ is less than or equal to zero. So this is the graph of all the things π₯ can be.

What is the greatest value π₯ could be? Zero. Every other value for π₯ is negative. Now, we can move on to checking the expression π₯ plus eight. If π₯βs largest possible value is zero, then the largest value of π₯ plus eight is eight.

Given these limits for π₯, the greatest possible value of π₯ plus eight equals eight.