in this question we'll take a look at four basic absolute value inequalities and we'll graph the solution set and also Express the solution set using interval notation remember the absy value of a real number is a real number's distance from zero so looking at our first inequality we have the absolute value of x is less than three so we're looking for all the numbers on the number line whose distance from zero is less than three units and that would be the numbers between positive3 and neg3 and notice in this case the end points would not be included because 3 andg -3 are exactly three units from zero not less than three units so a couple ways to show this graphically using points we'd make an open Point Ong -3 an open point on positive3 and then we graph the interval between these two values these are all the real numbers on the number line whose distance from zero is less than or equal to three units another way to graph the same interval would be instead of using open points we use parentheses we have an open parenthesis on Nega -3 open parenthesis on positive three and graph between the values these graphs indicate the exact same interval using interval notation we have the interval from -3 to positive3 but because the end points are not included we use an open parenthesis here and here if the end points were included we use square brackets not parentheses next we have the absite value of x less than or equal to 6 so the solution set consists of all the real numbers that're less than or equal to 6 units from zero so all the values between 6 and -6 this time including the endpoints would satisfy this inequality so we make a Clos point on -6 a Clos point on positive 6 and we graph all the values between I forgot if I mentioned before but using our online homework the open Dot means means the open point and the dot means the Clos Point another way to graph the same interval would be to use square brackets on the number line indicating the end points are included in the interval so this interval is the exact same interval using different notation which looks very similar to interval notation so the solution set interval using interval notation is from -6 to positive 6 the end points are included so we use square brackets this is called a closed interval because the end points are included this is called an open interval because the end points are not included let's take a look at two more examples here we have negative absolute value of x greater than or equal to -6 the first step here is to solve the inequality for the absolute value of x to do this we have to multiply or divide both sides of the equation by -1 but we know when we do this to solve an inequality we must reverse the inequality symbol so if we multiply both sides by 1 we'd have the absolute value of x is less than or equal to POS 6 so the solution set or interval would be all the real numbers on the number line whose distance from zero are less than or equal to 6 units so now this is just like the previous example we have closed points on -6 and six and we graph the interval between the values or we could also use square brackets so interval notation again the same as our last question from -6 to positive 6 the end points are included so we have square brackets here and now for our last example we have 2 * the ABS I of X less than 10 again our goal here is to First solve for the ABS Val of X so we would divide both sides by two since we're dividing by a positive we do not reverse the inequality symbol so we have the absolute V of X less less than five so we're looking for all the real numbers on the number line whose distance from zero is less than five units not equal to five units but less than five units which means in this case the values would be between neg5 and five not including the endpoints so open Point open point and all the values between these two or we could also use a ROM parentheses here and here indicating the same interval so the interval is from 5 to 5 using interval notation the end points are not included so we don't use square brackets we use a Randon parenthesis here and here I hope you found this helpful