1. **State the problem:** Differentiate between rational and irrational numbers. 2. **Explanation:** - Rational numbers are numbers that can be expressed as a fraction of two integers, i.e., $\frac{a}{b}$ where $a,b \in \mathbb{Z}$ and $b \neq 0$. - Irrational numbers cannot be expressed as such fractions; their decimal expansions are non-terminating and non-repeating. 3. **State the problem:** Show that $\sqrt{2}$ is an irrational number. 4. **Proof by contradiction:** - Assume $\sqrt{2}$ is rational, so $\sqrt{2} = \frac{a}{b}$ where $a,b$ are coprime integers. - Square both sides: $2 = \frac{a^2}{b^2}$ so $a^2 = 2b^2$. - Thus $a^2$ is even, implying $a$ is even. - Let $a = 2k$, then $a^2 = 4k^2 = 2 b^2 \Rightarrow b^2 = 2k^2$ so $b^2$ is even, implying $b$ is even. - This contradicts that $a,b$ are coprime since both are even. - Hence, $\sqrt{2}$ is irrational. 5. **State the problem:** Simplify $\frac{3+\sqrt{24}}{2+\sqrt{6}}$ by rationalizing the denominator. 6. **Work:** - Multiply numerator and denominator by the conjugate of denominator $2-\sqrt{6}$: $$\frac{(3+\sqrt{24})(2-\sqrt{6})}{(2+\sqrt{6})(2-\sqrt{6})}$$ - Denominator: $2^2 - (\sqrt{6})^2 = 4 - 6 = -2$ - Numerator: $(3)(2) - 3\sqrt{6} + 2\sqrt{24} - \sqrt{24} \sqrt{6}$ - Simplify each term: - $3 \times 2 = 6$ - $-3\sqrt{6}$ stays as is - $2\sqrt{24} = 2 \times 2\sqrt{6} = 4\sqrt{6}$ since $\sqrt{24} = 2\sqrt{6}$ - $\sqrt{24} \sqrt{6} = \sqrt{144} = 12$ - Numerator: $6 - 3\sqrt{6} + 4\sqrt{6} - 12 = (6-12) + (-3\sqrt{6} + 4\sqrt{6}) = -6 + \sqrt{6}$ - Final expression: $$\frac{-6 + \sqrt{6}}{-2} = \frac{-6}{-2} + \frac{\sqrt{6}}{-2} = 3 - \frac{\sqrt{6}}{2}$$ 7. **State the problem:** Solve for $x$ in $\frac{5x+2}{4} - \frac{3}{2} = \frac{7x-1}{3}$ 8. **Work:** - Multiply both sides by 12 (LCM of denominators 4,2,3) to clear fractions: $$12 \times \left(\frac{5x+2}{4} - \frac{3}{2}\right) = 12 \times \frac{7x-1}{3}$$ - Simplify: $$3(5x+2) - 6(3) = 4(7x-1)$$ - Expand: $$15x + 6 - 18 = 28x - 4$$ - Simplify: $$15x - 12 = 28x - 4$$ - Move terms: $$-12 + 4 = 28x - 15x$$ $$-8 = 13x$$ - Solve for $x$: $$x = -\frac{8}{13}$$