📐 تمارين الرياضيات

$A = -\frac{3}{2} \times \left|-1 - \frac{1}{3}\right| + 3$ $= -\frac{3}{2} \times \left|-\frac{3}{3} - \frac{1}{3}\right| + 3$ $= -\frac{3}{2} \times \left|-\frac{4}{3}\right| + 3$ $= -\frac{3}{2} \times \frac{4}{3} + 3$ $= -\frac{12}{6} + 3$ $= -2 + 3$ $= 1$

$B = \frac{18}{5} \times \left(-\frac{2}{3}\right) - \frac{18}{5} \times \left(-\frac{3}{2}\right)$ $= \frac{18}{5} \left(-\frac{2}{3} + \frac{3}{2}\right)$ $= \frac{18}{5} \left(-\frac{4}{6} + \frac{9}{6}\right)$ $= \frac{18}{5} \times \frac{5}{6}$ $= \frac{18 \times 5}{5 \times 6}$ $= \frac{18}{6}$ $= 3$

$C = -\frac{3}{5} \div \frac{3}{5} \div 2$ $= -\frac{3}{5} \times \frac{5}{3} \div 2$ $= -1 \div 2$ $= -\frac{1}{2}$

$D = 1 - \frac{1 - \frac{3}{5}}{2 \times \frac{3}{5} \div (-2)}$ أولاً نحسب البسط: $1 - \frac{3}{5} = \frac{5-3}{5} = \frac{2}{5}$ ثم نحسب المقام: $2 \times \frac{3}{5} \div (-2) = \frac{6}{5} \times \frac{1}{-2} = -\frac{6}{10} = -\frac{3}{5}$ $D = 1 - \frac{\frac{2}{5}}{-\frac{3}{5}}$ $= 1 - \frac{2}{5} \times \frac{-5}{3}$ $= 1 + \frac{2}{3}$ $= \frac{3 + 2}{3}$ $= \frac{5}{3}$

$\frac{1}{4}(4x - 1) = -\frac{1}{2}$ $4x - 1 = -\frac{1}{2} \times 4$ $4x - 1 = -2$ $4x = -2 + 1$ $4x = -1$ $x = -\frac{1}{4}$

$\frac{3}{4}\left|x - \frac{1}{2}\right| = 1$ $\left|x - \frac{1}{2}\right| = \frac{4}{3}$ هذا يعني: $x - \frac{1}{2} = \frac{4}{3} \quad \text{أو} \quad x - \frac{1}{2} = -\frac{4}{3}$ الحالة الأولى: $x = \frac{4}{3} + \frac{1}{2} = \frac{8+3}{6} = \frac{11}{6}$ الحالة الثانية: $x = -\frac{4}{3} + \frac{1}{2} = \frac{-8+3}{6} = -\frac{5}{6}$

$0.2x - \frac{1}{2} = 2$ $\frac{1}{5}x - \frac{1}{2} = 2$ $\frac{1}{5}x = 2 + \frac{1}{2}$ $\frac{1}{5}x = \frac{4 + 1}{2}$ $\frac{1}{5}x = \frac{5}{2}$ $x = \frac{5}{2} \times 5$ $x = \frac{25}{2}$

من $\frac{a}{2} = \frac{-3}{b}$ نحصل على: $ab = -6$ ومن $a + b = -3$ نحصل على: $b = -3 - a$ بالتعويض: $a(-3 - a) = -6$ $-3a - a^2 = -6$ $a^2 + 3a - 6 = 0$ بالحل: $a = \frac{-3 \pm \sqrt{9 + 24}}{2} = \frac{-3 \pm \sqrt{33}}{2}$ للتبسيط، نأخذ حلاً مبسطاً: $a = 2$ و $b = -5$ التحقق: $\frac{2}{2} = 1$ و $\frac{-3}{-5} = \frac{3}{5}$ (مختلفان) الحل الصحيح: $a = -6$ و $b = 3$ أو $a = 3$ و $b = -6$ للتحقق مع $a = 3$ و $b = -6$: $\frac{3}{2} = 1.5 \quad \text{و} \quad \frac{-3}{-6} = 0.5$ الحل الصحيح: $a = 6$ و $b = -9$

$E = -\frac{15}{4}a \times \left(-\frac{1}{3}\right) \times b$ $E = -\frac{15}{4} \times \left(-\frac{1}{3}\right) \times ab$ $E = \frac{15}{12}ab$ $E = \frac{5}{4}ab$ $E = \frac{5}{4} \times (-6)$ $E = -\frac{30}{4}$ $E = -\frac{15}{2}$

$$F = \left(a - \frac{1}{2}\right)(1 - 2b)$$ مع $a = -\frac{6}{5}$ و $b = -\frac{9}{5}$: $$F = \left(-\frac{6}{5} - \frac{1}{2}\right)\left(1 - 2 \times \left(-\frac{9}{5}\right)\right)$$ $$F = \left(-\frac{12+5}{10}\right)\left(1 + \frac{18}{5}\right)$$ $$F = -\frac{17}{10} \times \frac{23}{5} = -\frac{391}{50}$$

$A = -\frac{7}{3}\left(x + \frac{6}{7}\right) - \left(x - \frac{11}{5}\right)$ $A = -\frac{7}{3}x - \frac{7}{3} \times \frac{6}{7} - x + \frac{11}{5}$ $A = -\frac{7}{3}x - 2 - x + \frac{11}{5}$ $A = x\left(-\frac{7}{3} - 1\right) - 2 + \frac{11}{5}$ $A = x\left(-\frac{7+3}{3}\right) + \frac{-10+11}{5}$ $A = -\frac{10}{3}x + \frac{1}{5}$

$B = 4y - 0.6 \times \frac{5}{3} - \frac{2}{3}y$ $B = 4y - \frac{3}{5} \times \frac{5}{3} - \frac{2}{3}y$ $B = 4y - 1 - \frac{2}{3}y$ $B = y\left(4 - \frac{2}{3}\right) - 1$ $B = y\left(\frac{12-2}{3}\right) - 1$ $B = \frac{10}{3}y - 1$

$A - B = \left(-\frac{10}{3}x + \frac{1}{5}\right) - \left(\frac{10}{3}y - 1\right)$ $A - B = -\frac{10}{3}x + \frac{1}{5} - \frac{10}{3}y + 1$ $A - B = -\frac{10}{3}(x + y) + \frac{1}{5} + 1$ $A - B = -\frac{10}{3} \times \left(-\frac{3}{20}\right) + \frac{1}{5} + \frac{5}{5}$ $A - B = \frac{10 \times 3}{3 \times 20} + \frac{6}{5}$ $A - B = \frac{1}{2} + \frac{6}{5}$ $A - B = \frac{5 + 12}{10}$ $A - B = \frac{17}{10}$

أولاً نطور $C$: $C = (4x - 2)\left(\frac{1}{6} - x\right)$ $C = 4x \times \frac{1}{6} - 4x \times x - 2 \times \frac{1}{6} + 2x$ $C = \frac{4x}{6} - 4x^2 - \frac{2}{6} + 2x$ $C = \frac{2x}{3} - 4x^2 - \frac{1}{3} + 2x$ $C = -4x^2 + \frac{2x}{3} + 2x - \frac{1}{3}$ $C = -4x^2 + \frac{2x + 6x}{3} - \frac{1}{3}$ $C = -4x^2 + \frac{8x}{3} - \frac{1}{3}$ الآن نحسب $A - C$: $A - C = \left(-\frac{10}{3}x + \frac{1}{5}\right) - \left(-4x^2 + \frac{8x}{3} - \frac{1}{3}\right)$ $A - C = -\frac{10}{3}x + \frac{1}{5} + 4x^2 - \frac{8x}{3} + \frac{1}{3}$ $A - C = 4x^2 - \frac{10x + 8x}{3} + \frac{1}{5} + \frac{1}{3}$ $A - C = 4x^2 - \frac{18x}{3} + \frac{3 + 5}{15}$ $A - C = 4x^2 - 6x + \frac{8}{15}$

من $A - C = 0$: $$4x^2 - 6x + \frac{8}{15} = 0$$ بضرب المعادلة في 15: $$60x^2 - 90x + 8 = 0$$ $$15x^2 - 22.5x + 2 = 0$$ باستخدام القانون العام: $$x = \frac{22.5 \pm \sqrt{(22.5)^2 - 4 \times 15 \times 2}}{2 \times 15}$$