In Exercises 95–102, use interval notation to represent all value...

Question

In Exercises 95–102, use interval notation to represent all values of x satisfying the given conditions.

y = 8 - |5x + 3| and y is at least 6

Accepted Answer

Hello everyone in this video we're gonna be looking at this practice problem where we're trying to find the values of X that satisfy the conditions for why being greater than seven and in this case why is equal to one plus the absolute value of two minus X. And we want to make sure to express that range and interval notation. So in this case you can see that why being greater than seven is the range of values that we want. So if I take why greater than seven I can take this substitution where y. Is equal to one plus the absolute value of two minus X. And I'm going to plug that in to my why? So this becomes one plus the absolute value of two minus X. Greater than seven. And now I have an inequality with X. For the values that I want. So now I can solve this inequality and this will give me the range of X values that satisfy why being greater than seven. So if I solve this I subtract one from both sides will have the absolute value of 2 -1 greater than six. And recall that when we have an absolute value we can either have a positive on the inside or we can have a negative. So I'm gonna solve for both of those scenarios where now I have two minus X greater than six and negative times two minus X. Greater than six. So this one I can just subtract two and I get negative X greater than negative or positive. For advice, attract a negative across inequality. The inequality switches signs. So now it's less than I have negative four. So x less than -4. And if I solve on my right hand side or for my negative value Again I have to divide by negative. So I have to switch the inequality so greater than changes to last then and I have negative six and I have 2 -1. Can I subtract 2? I have negative X Less than negative eight and I have to divide by that negative again. So the inequality switches signs again. So I'm off with X greater than positive eight. So now I have to range of values to represent my exes, I have X greater than eight and X less than negative four. So in this case if I were to draw a number line, not really to scale. So we have negative four here, zero here and eight up here. So X less than negative force saying that I want the range of values down this way. Notice how it's not an equal to So this would be of parentheses and I want them all the way down. So there's no limit As long as they're less than -4. So going all the way to negative infinity And then I have X greater than eight. So in this case I just wanted to be greater than eight notice there's no upper limit either. So this seems to be going to positive infinity and again a parentheses because it is not equal to. So this gives me my range of X values that will result in why being greater than seven. So if I write this out in an interval notation, I have negative infinity to negative four parentheses and I have a union eight to positive infinity. And this becomes my final answer. And you can see that answer choice D also has negative infinity to negative four, union with eight to positive infinity. So hopefully this video helped and see you next time.

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