> [!summary] This proof uses the definition of the absolute value in different forms to prove the triangle inequality. > **Key Result:** > $|a+b| \leq |a| + |b|$ >[!info]+ Read Time **⏱ 1 min** # Mathematical Proof > [!warning] Assumptions To prove the absolute value multiplication rule by [[Proof by Cases|proof by cases]] assume the following: > - The definition of the absolute value of a $|a+b|$ is: > $ > | a+b| = > \begin{cases} > a+b, & \text{if } a+b \geq 0 \\ > -(a+b), & \text{if } a+b < 0 > \end{cases} > $ > - The definition of an absolute value of $|a|$ is: > $ > |a| = > \begin{cases} > a, & \text{if } a \geq 0 \\ > -a, & \text{if } a < 0 > \end{cases} > $ > - The definition of an absolute value of $|b|$ is: > $ > |b| = > \begin{cases} > b, & \text{if } b \geq 0 \\ > -b, & \text{if } b < 0 > \end{cases} > $ Prove that $|a+b| \leq |a| +|b|$ $ \begin{array}{c} \text{Case 1:} \quad (a+b) \geq 0: \\ a+b \leq |a| +|b| \quad \text{By defintion of the abs value of a,b} \\ \\ \text{Case 2:} \quad (a+b)<0: \\ \text{This is saying $-(a+b)$ by definiton of abs value of a+b}\\ -a + -b \leq |a| + |b| \quad \text{By definition of the abs value of a,b} \end{array} $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/051WWg9twO0?si=tP3UQdOcerjiEoyR" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> --- <!-- Light Mode Newsletter Embed --> <div class="mm-form-light"> <iframe src="https://updates.cyberleadhub.com/widget/form/Y0kpQVpjJQuxEfX59m17" id="inline-Y0kpQVpjJQuxEfX59m17" title="Join Math & Matter Newsletter (Light)" data-height="900" scrolling="no" allowtransparency="true" loading="lazy" style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden" ></iframe> </div> <!-- Dark Mode Newsletter Embed --> <div class="mm-form-dark"> <iframe src="https://updates.cyberleadhub.com/widget/form/lbeDLm24VjuaFxhjccA1" id="inline-lbeDLm24VjuaFxhjccA1" title="Join Math & Matter Newsletter (Dark)" data-height="900" scrolling="no" allowtransparency="true" loading="lazy" style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden" ></iframe> </div> <!-- Provider script (only once) --> <script src="https://updates.cyberleadhub.com/js/form_embed.js"></script> ---