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

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

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

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

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

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

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

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

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

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

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