# 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 $x$.

Please explain in detail.

• Did you try a web search? Google gives this as the second hit for "rules of inequalities". Mar 16, 2012 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. Mar 16, 2012 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 $\leq \ \leftrightarrow \ \geq$ Mar 16, 2012 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. Mar 16, 2012 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! Mar 16, 2012 at 14:48

$$-8 \leq 1 - 3(x-2) \leq 13$$ $$-8 -1 \leq 1 -1 - 3(x-2) \leq 13-1 \implies -9 \leq - 3(x-2) \leq 12$$ $$9 \geq 3(x-2) \geq -12 \implies 9 \geq 3(x-2) \geq -12 \implies 3 \geq (x-2) \geq -4$$ $$3+2 \geq x-2+2 \geq -4+2 \implies 5 \geq x \geq -2$$
• 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? Mar 16, 2012 at 14:30