An Introduction to Inequalities
Absolute Value Equations
Equations vs Inequalities
- An equation is a statement where two expressions are equal (=) to each other.
- An inequality, on the other hand, is a statement where two expressions are unequal.
- There are four types of inequalities we will work on within this course:
- less than (<)
- greater than (>)
- less than or equal to (≤)
- greater than or equal to (≥).
Absolute Value Equations
Equations vs Inequalities
- A number line shows relationships between real numbers.
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Division of Radicals
Division of Radicals
Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{54}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)
=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{9\sqrt[3]{2}}{4}\)
Division of Radicals
Division of Radicals
Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{54}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)
=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{9\sqrt[3]{2}}{4}\)
Division of Radicals
Division of Radicals
Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{54}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)
=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{9\sqrt[3]{2}}{4}\)
Division of Radicals
Division of Radicals
Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{54}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)
=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{9\sqrt[3]{2}}{4}\)
Division of Radicals
Division of Radicals
Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{54}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)
=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{9\sqrt[3]{2}}{4}\)
Division of Radicals
Division of Radicals
Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{54}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)
=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{9\sqrt[3]{2}}{4}\)
Division of Radicals
Division of Radicals
Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{54}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)
=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{9\sqrt[3]{2}}{4}\)
Division of Radicals
Division of Radicals
Ex 1: Simplify \(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
\(\frac{15 \sqrt[3]{108}}{20 \sqrt[3]{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{\frac{108}{2}}\)
=\(\frac{15}{20} \cdot \sqrt[3]{54}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27 \cdot 2}\)
=\(\frac{15}{20} \cdot \sqrt[3]{27} \cdot \sqrt[3]{2}\)
=\(\frac{15}{20} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{\cancel{15}^{~3}}{\cancel{20}^{~4}} \cdot 3 \cdot \sqrt[3]{2}\)
=\(\frac{9\sqrt[3]{2}}{4}\)
Division of Radicals
Division of Radicals
Ex 2: Simplify \(\frac{\sqrt{6}}{\sqrt{5}}\)
\(\frac{\sqrt{6}}{\sqrt{5}}\)
=\(\frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}\)
=\(\frac{\sqrt{6} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}\)
=\(\frac{\sqrt{6 \cdot 5}}{5}\)
=\(\frac{\sqrt{30}}{5}\)
Division of Radicals
Division of Radicals
Ex 3: Simplify \(\frac{\sqrt{6}}{\sqrt{5}}\)
\(\frac{\sqrt{6}}{\sqrt{5}}\)
=\(\frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}\)
=\(\frac{\sqrt{6} \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}\)
=\(\frac{\sqrt{6 \cdot 5}}{5}\)
=\(\frac{\sqrt{30}}{5}\)
\pi \cdot \pi
An Introduction to Inequalities
By Anurag Katyal
