 So let's try to solve the absolute value inequality absolute value of minus 2x minus 1 must be greater than 3 So we solve inequalities by solving the corresponding equality absolute value minus 2x minus 1 equal to 3 So remember the absolute value equation becomes two equations minus 2x minus 1 equals 3 and minus 2x minus 1 equals negative 3 So we'll solve minus 2x minus 1 equals 3 first. We'll isolate our x terms by adding one to both sides That gives us minus 2x equals 4 and since the left-hand side is the product minus 2 times x we can solve for x by dividing by the coefficient minus 2 and Simplifying the algebra the factors of minus 2 can be removed and for divided by minus 2 is Minus 2 and that gives us our first solution x equals negative 2 The second equation we can solve in the same way minus 2x minus 1 equals negative 3 We have a subtraction over on the left-hand side So we can add plus 1 to get rid of the subtraction once again We have a product over on the left-hand side minus 2 times x said to get rid of a product will divide And we get our solution x equals 1 which tells us the critical values are x equals minus 2 and x equals 1 Now if we want to find the solution set we can begin by graphing the critical values Now since the critical values solve the equality minus 2x minus 1 equal to 3 But our inequality is strictly greater than the critical values are not included in the solution set So we'll use open circles for them So the critical values partition the number line into three intervals And now we'll pick a point in each interval and see if the points in the interval satisfy the inequality So in this interval on the left we might let x equal minus 1 million remember it's easier to go big and It's useful to think about these in terms of our signs and our magnitude So the components of our absolute value expression minus 2x Since x is negative and large then minus 2x will be Still large, but because we're multiplying two negative numbers. We'll get a positive number If we then subtract one from a large and positive number we get a number that is Still large and still positive and when we take the absolute value It's still large and still positive And now the question you got to ask yourself is is a large positive number greater than three and the answer is Yes, so x equals minus 1 million should be included in the solution set And we want to include the interval that it's part of so we'll shade this left-hand interval How about this central interval well unfortunately? We can't pick numbers that are large because they're way off to the right or way off to the left But there is a convenient number right here in the middle of the interval x equals zero So let's actually substitute x equals zero into our inequality and see if it's true So first we'll let x equals zero will substitute that into our absolute value inequality We'll do some arithmetic minus two times zero minus one is The absolute value of negative one is and the question we have to ask ourselves is is one greater than three And even though some people may think that one is bigger than three This is in fact false and so we should exclude this central interval And finally we have this interval off to the right. So let x be one Billion and so remember x is a large Positive number and let's go through the different components of our inequality. So minus 2x That's minus 2 times a large positive number. And so that's going to give us a number that is large and negative Minus 2x minus one well, I'm going to subtract one from a large negative number and I get a Large negative number, but then I take the absolute value and I get a Large positive number and again the question you got to ask yourself is is a large Positive number going to be greater than three And we won't listen to this critter We'll say that yes a large positive number is greater than three And so that means that x equals one billion should be in our solution set And so we should include the interval that contains x equals one billion Now once we have the interval we can write our answer in interval notation So this left interval goes from the way way way way way way left Well, we call that minus infinity and all the way up until we hit negative two and we don't include it So remember infinity is never included in our interval it always gets a parenthesis Negative two because it has an open circle shouldn't be included either And so we'll close parenthesis there and this describes our left interval minus infinity up to negative two We don't include the center portion But we do want to include this right hand interval which starts at one and goes to the way way way way way Right, which we call infinity and again infinity always gets a parenthesis One is not included in our interval. So one also gets a parenthesis Since our domain includes things that are in the one or in the other will union these two intervals And to avoid confusion we'll say that x is an element of the union of these two intervals