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

5 |4...

Question

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

5 > |4 - x|

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we want to simplify the inequality to greater than the absolute value of 5 -7 x. And when you want to express the solution set for the inequality in interval notation. So in this case you can see that we're working with an absolute value and recall that whenever you have an absolute value, the number inside is always expressed as positive even if it's negative. So in this case we want to solve for those two scenarios where if the number inside was negative, we would have a different inequality than if the number inside was positive. So if I look at my Negative I would have to greater than -5 -7 x. And if I had a positive number I would have to greater than positive 5 -7 x. So looking at this negative side first, if I divide the negative and left with negative two, I have to switch the sign because I divided it a negative across an inequality Leaving me with -2 less than five minus seven X. And my positive number would simply be too Greater than 5 -7 x. So you can see that the difference between if the number were negative or if the number were positive is that there is a sign switch and the two has also switched signs. So I'm gonna go ahead and solve for both of these inequalities to try to get the range of X values that will satisfy The condition to greater than the absolute value of 5 -7 x. So starting with negative two less than five minus seven X subtract five, leaving me with negative seven less than negative seven X divide by negative seven. Notice how I'm dividing by negative across an inequality. So my science which is again and I'm left with one greater than X. I can rearrange that so x less than one and go go ahead and simplify to greater than five minus seven X. Also if I do that I'm gonna subtract five Left with -3 greater than -7 x divided by -7. Notice I'm dividing by a negative once again. So I'm switching the sign so that leaves me with three sevens less than X or X is greater than 3/7. Now I have These two inequalities that give me the range of X values that will satisfy the condition or the inequality of two greater than the absolute value of 5 -7 x. If we look on a number line, Say this is zero 3/ and this is one my first Interval or inequality seeing X has to be less than one. So X has to be less than one and my second interval Or inequality X greater than 3/7 the same X has to be greater than 3/7. So this way and you can see that the two places where those ranges overlap is three sevens to one. So this becomes the final range of values where X satisfies the condition To greater than absolute value 5 -7 x. And you can see 3/7 to parentheses is expressed in interval notation. So this becomes my final answer, and this aligns with answer choice D. So hopefully this video helped and see you next time.

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