# Inequalities and Absolue Values: Problem Solving Approach

Tags: absolue, inequalities, solving, values
 P: 3,016 Hello all! In my quest to re-teach myself the basics of mathematics in a more rigorous fashion, I have found out that inequalities and absolute values are a weak point if mine. So I am working to address that. I am getting much better at it (with help from PF), but I have recently encountered seemingly simple problem that turned out to be a little trickier than I thought. Though I can arrive at the correct answer, I am not sure that my procedure is sound. Hopefully you can offer some insight. Take the following problem from chapter 1 of Spivak's Calculus (Problem 11 (iv)): Find all ##x## for which ##|x-1|+|x-2| > 1 \qquad(1)##. My approach to these has been to use the fact that the definition of absolute value is \begin{align} |x| = \begin{cases} x, & \text{if }x\ge0 \\ -x, & \text{if }x\le0 \end{cases} \end{align} so then for each quantity enclosed by absolute value signs, there are 2 cases that needed to be evaluated. Applying this to (1) we have Case 1: ##(x-1)>0 \wedge (x-2)>0## then ##(x-1) + (x-2) > 1 \implies x > 2.## Case 2: ##(x-1)<0 \wedge (x-2)<0## then ## (1-x) + (2-x) > 1 \implies x<2.## Case 3: ##(x-1)>0 \wedge (x-2)<0## then ## (x-1)+(1-x) > 1 \implies 0 >1. ## Case 4: ##(x-1)<0 \wedge (x-2)>0## then ## (1-x) + (x+2) > 1 \implies 3 > 1. ## Let's just look at Case 1 for a moment. Assuming that ##(x-1)>0 \wedge (x-2)>0## is the same as assuming ## x > 1 \wedge x>2.## This is clearly only true for ##x>2##, so there is really no need to specify that ##x>1.## But when it comes time to solve the actual problem, I need to use the expression ##(x-1)## under the assumption that it is a positive quantity, which is the same as specifying that ##x>1##. The answer I got is ##x>2## and is valid, but I feel like I might miss something in future problems if I do not pay attention to this detail. So my question is this: Do I simply solve the inequality as I have done and then restrict the solution to ##x>2## if I were to get something less than 2? Does my question make sense?
 Mentor P: 18,346 Inequalities and Absolue Values: Problem Solving Approach The case $x>1~\wedge~x>2$ is indeed equivalent to x>2. If you solve the equation under the premisse that x>2, then every solution must satisfy that. If you find that the equation is true for all x>-2, then only the x>2 will count. For example, if you solve $|x-1|>-5$ (I know you can easily see that all x will be a solution, but I'm just setting an example). You can split up 1) $x\geq 1$, in that case |x-1|=x-1. So the equation is x-1>-5. This is true for x>-4. However, you originally set $x>1$, so in this case the only solutions are all $x>1$ (and not all x>-4). 2) x<1, in that case |x-1|=1-x. The equation becomes 1-x>5, or x<6. In this case the solutions are all x<1 (and not x<6). Adding (1) and (2) yields that all real numbers are a solution.