# How to solve Linear Inequalities? What are the rules that govern to solve them and why they are there?

Given: $-8 \leq 1-3(x-2) \leq 13$. How to solve this inequality? What rules apply here. I meant to ask when do we have to replace the $\leq$ sign with $\geq$ sign and why?

What are the rules that govern to solve them and why they are there? I just want to take the above inequality as a typical scenario to dicscuss what is happening and why, while solving it for $x$.

-
Did you try a web search? Google gives this as the second hit for "rules of inequalities". – The Chaz 2.0 Mar 16 '12 at 13:53
Yes i have. There are resources available online but i like to discuss these rules particularly when it somes to replacing <= with >=. Thanks for the link. – Maxood Mar 16 '12 at 14:12
"Discuss" - ??? This site facilitates the "Question and Answer" model more than ongoing discussion... (see the FAQ above). The rule is that multiplication by a negative constant changes (all of) the signs $\leq \ \leftrightarrow \ \geq$ – The Chaz 2.0 Mar 16 '12 at 14:17
I have read the rules and like to particcipate in the discussion. It does say that questions of any level can be asked.I also like to visualize the simplified inequality in terms of intervals or graphs. Let me know if i am wrong here. – Maxood Mar 16 '12 at 14:27
Questions of any level can indeed be asked. You have asked three questions (Which I will re-order and reword): "What is the process for solving such an inequality?" "When and why do we change the direction of inequality?" "What is a worked solution?" - If you researched this topic, the first two would have been answered. Then you would have been able to attempt your own solution. Many times I do not represent more than one user in the community. Who knows - this might be one of those times! – The Chaz 2.0 Mar 16 '12 at 14:48

Let's consider what multiplication by -1 means geometrically. Imagine a number. You are at the number 5. Multiply by -1. You are now at -5 and geometrically we see that multiplication by -1 means to reflect about 0.

Now reflecting about 0 preserves the distance from zero. This is why |-5| and |5| are the same number. Reflection about zero is a distance preserving action.

But inequality doesn't measure distance from zero. Inequality measure which number is further to the right on the number line. This last fact must be kept in mind. Now, when I take two points on the number line, say A and B, and the reflect about zero what happens? The point that is furthest right reflects to a point that is to the left of the reflection of the point that was furthest left.

So when solving inequalities one must reverse the inequality when multiplying by a negative number. Multiplying by a negative number reverses the order of the numbers.

-
Excellent explanation. Thankyou so much for your clear words. Ireally appreciate it. – Maxood Mar 16 '12 at 14:29
You're welcome. – YequalsX Mar 16 '12 at 15:37

$$-8 \leq 1 - 3(x-2) \leq 13$$ $$-8 -1 \leq 1 -1 - 3(x-2) \leq 13-1 \implies -9 \leq - 3(x-2) \leq 12$$ $$9 \geq 3(x-2) \geq -12 \implies 9 \geq 3(x-2) \geq -12 \implies 3 \geq (x-2) \geq -4$$ $$3+2 \geq x-2+2 \geq -4+2 \implies 5 \geq x \geq -2$$

-
The solution above is quite logical enough to understand what is happening. So basically we >= with <= when we multiply both sides by -1...right? Also how 5 >= x >= -2 do we represent them intervals or grpah? Thanks – Maxood Mar 16 '12 at 14:21
Yes. $5 \geq x \geq -2$ can be represented by all points on the number line between 5 an -2. Just a friendly suggestion: May I suggest you read this for asking questions with better formatting? – Inquest Mar 16 '12 at 14:30
I see. Thanks for the link. – Maxood Mar 16 '12 at 14:34