The uploaded document contains four mathematical problems related to solving equations and inequalities. Let's solve each one step by step.

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Exercice 1

Solve the equation: $\frac{7}{x+2} = \frac{3}{2x-1}$

Cross-multiply to eliminate the fractions: $7(2x - 1) = 3(x + 2)$

Expand and simplify: $14x - 7 = 3x + 6$

Bring all terms involving $x$ to one side and constants to the other: $14x - 3x = 6 + 7$ $11x = 13$

Solve for $x$: $x = \frac{13}{11}$

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Exercice 2

Solve the inequality: $(2x + 6)(x - 4) \geq 2(x + 7)^2$

Expand both sides: $(2x + 6)(x - 4) = 2x^2 - 8x + 6x - 24 = 2x^2 - 2x - 24$ $2(x + 7)^2 = 2(x^2 + 14x + 49) = 2x^2 + 28x + 98$

The inequality becomes: $2x^2 - 2x - 24 \geq 2x^2 + 28x + 98$

Simplify by canceling $2x^2$ and rearranging: $-2x - 24 \geq 28x + 98$ $-2x - 28x \geq 98 + 24$ $-30x \geq 122$

Solve for $x$ (reverse inequality because dividing by negative): $x \leq -\frac{122}{30} = -\frac{61}{15}$

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Exercice 3

Solve the equation: $9x^2 - 16 - (6x - 8)(x + 3) = 0$

Expand $(6x - 8)(x + 3)$: $(6x - 8)(x + 3) = 6x^2 + 18x - 8x - 24 = 6x^2 + 10x - 24$

Substitute into the equation: $9x^2 - 16 - (6x^2 + 10x - 24) = 0$

Simplify: $9x^2 - 16 - 6x^2 - 10x + 24 = 0$ $3x^2 - 10x + 8 = 0$

Factorize or use the quadratic formula: $x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(8)}}{2(3)}$ $x = \frac{10 \pm \sqrt{100 - 96}}{6}$ $x = \frac{10 \pm 2}{6}$

Solve: $x = \frac{12}{6} = 2 \quad \text{or} \quad x = \frac{8}{6} = \frac{4}{3}$

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Exercice 4

Solve the equation: $\frac{6x^2 - 1}{(-x - 8)(2x + 1)} = -3$

Cross-multiply: $6x^2 - 1 = -3(-x - 8)(2x + 1)$

Expand $(-x - 8)(2x + 1)$: $(-x - 8)(2x + 1) = -2x^2 - x - 16x - 8 = -2x^2 - 17x - 8$

Substitute: $6x^2 - 1 = -3(-2x^2 - 17x - 8)$ $6x^2 - 1 = 6x^2 + 51x + 24$

Simplify: $-1 = 51x + 24$ $51x = -25$ $x = -\frac{25}{51}$

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Related Questions:

1. How do you solve rational equations with multiple variables?
2. What are common methods to simplify quadratic inequalities?
3. Can we use graphing to solve these problems? If yes, how?
4. How does cross-multiplication work in rational equations?
5. What are real-life applications of solving equations and inequalities?

Tip:

Always check solutions of equations and inequalities in the original problem to ensure no restrictions are violated (e.g., division by zero).