In Exercises 59–94, solve each absolute value inequality.12

Question

In Exercises 59–94, solve each absolute value inequality.12 < |- 2x + 6/7| + 3/7

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we want to simplify the inequality five less than the absolute value of negative three X plus five minus two and we want to express our answer in interval notation. So in this case you can see that we're dealing with an absolute value inequality which means we have to account for the fact that in an absolute value this number could either be positive or negative. So I'm going to go ahead and solve the inequality twice. One for if the number was positive and another fourth the number inside the absolute value was negative. So if I look at if the number was positive I have five less than positive negative three X plus five minus two. And if the number was negative I would have five less than negative parentheses negative three X plus five minus two. So I'm gonna go ahead and simplify if the number was positive first. So I'm going to add to I have seven less than negative three X plus five. Notice how distributing this positive didn't change the signs of the numbers on the inside and then I'm gonna subtract five. So I have two less than negative three X. And I'm going to divide by negative three. And if I divide by -3 I have to switch the sign because I'm dividing by negative across inequality. So that leaves me with negative 2/3 greater than X. And if I rearrange this than I have X is less than -2/3. So that is the range of X values that I get when I simplify if the number inside the absolute value was positive. So I'm gonna go ahead and simplify if the number was negative. Now Following the same process as before I'm Gonna Add two. I have seven less than negative negative three X plus five. If I distribute this negative I have seven less than negative times negative three X. I love with positive three X. And negative times positive five. And left with negative five. and I'm gonna add 5 to both sides. So that leaves me with 12 less than three x. and five x 3. I have four less than X. I'm going to rearrange this And that leaves me with X greater than four. So this represents the range of values for X that satisfy this inequality. So I have these written in inequalities so let's go ahead and translate these inequalities into interval notation. So my first inequality is saying X is less than negative two thirds. So two thirds represents the upper bound or limit. And I'm gonna do parentheses because there is no equal to and there's no lower limit express. So I can write negative infinity as my lower limit. And you can see that my second inequality X greater than four is saying that or is the lower limit again parentheses because there is no equal to and there's no upper limit extra has to be greater than four. So I can write infinity as my upper limit. So now these represent the range of X values that satisfy the inequality and you can see that negative two thirds and four do not overlap. So I need to draw a union between my two intervals. So my final answer becomes negative infinity to to to negative 2/3 Union with to positive infinity. And this becomes the final range of X values that will satisfy the condition five less than absolute value of negative three X plus five minus two. And you can see that this aligns with answer choice D. So hopefully this video hoped and see you next time.

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In Exercises 59–94, solve each absolute value inequality.12 < |- 2x + 6/7| + 3/7

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