Devoir à domicile ISemestre II Exercice 1 \( \begin{array}{c}\text { 1) Résoudre les équations suivantes : } \\ 3 x-8=-2 \\ 3(2 x-5)-1=2(3 x-1) \\ \frac{2 x-1}{5}=\frac{x+5}{2}-x \\ x^{2}-5=0 \\ \text { 2) Résoudre les inéquations suivantes: } \\ 2 x-9 \geq 7\end{array} \)

Solve the inequality by following steps: - step0: Solve for \(x\): \(2x-9\geq 7\) - step1: Move the constant to the right side: \(2x\geq 7+9\) - step2: Add the numbers: \(2x\geq 16\) - step3: Divide both sides: \(\frac{2x}{2}\geq \frac{16}{2}\) - step4: Divide the numbers: \(x\geq 8\) Solve the equation \( \frac{2x-1}{5}=\frac{x+5}{2}-x \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{2x-1}{5}=\frac{x+5}{2}-x\) - step1: Multiply both sides of the equation by LCD: \(\frac{2x-1}{5}\times 10=\left(\frac{x+5}{2}-x\right)\times 10\) - step2: Simplify the equation: \(4x-2=-5x+25\) - step3: Move the expression to the left side: \(4x+5x=25+2\) - step4: Add and subtract: \(9x=25+2\) - step5: Add and subtract: \(9x=27\) - step6: Divide both sides: \(\frac{9x}{9}=\frac{27}{9}\) - step7: Divide the numbers: \(x=3\) Solve the equation \( 3(2x-5)-1=2(3x-1) \). Solve the equation by following steps: - step0: Solve for \(x\): \(3\left(2x-5\right)-1=2\left(3x-1\right)\) - step1: Calculate: \(6x-16=2\left(3x-1\right)\) - step2: Calculate: \(6x-16=6x-2\) - step3: Move the expression to the left side: \(6x-16-\left(6x-2\right)=0\) - step4: Calculate: \(-14=0\) - step5: The statement is false: \(x \in \varnothing \) Solve the equation \( 3x-8=-2 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3x-8=-2\) - step1: Move the constant to the right side: \(3x=-2+8\) - step2: Add the numbers: \(3x=6\) - step3: Divide both sides: \(\frac{3x}{3}=\frac{6}{3}\) - step4: Divide the numbers: \(x=2\) Solve the equation \( x^{2}-5=0 \). Solve the quadratic equation by following steps: - step0: Solve using square roots: \(x^{2}-5=0\) - step1: Add or subtract both sides: \(x^{2}=5\) - step2: Simplify the expression: \(x=\pm \sqrt{5}\) - step3: Separate into possible cases: \(\begin{align}&x=\sqrt{5}\\&x=-\sqrt{5}\end{align}\) - step4: Rewrite: \(x_{1}=-\sqrt{5},x_{2}=\sqrt{5}\) Voici les solutions pour les équations et inéquations données : ### 1) Résoudre les équations suivantes : 1. **Équation :** \( 3x - 8 = -2 \) **Solution :** \[ 3x - 8 = -2 \implies 3x = 6 \implies x = 2 \] 2. **Équation :** \( 3(2x - 5) - 1 = 2(3x - 1) \) **Solution :** \[ 3(2x - 5) - 1 = 2(3x - 1) \implies 6x - 15 - 1 = 6x - 2 \implies 6x - 16 = 6x - 2 \] Cette équation n'a pas de solution, donc : \[ x \in \varnothing \] 3. **Équation :** \( \frac{2x - 1}{5} = \frac{x + 5}{2} - x \) **Solution :** \[ \frac{2x - 1}{5} = \frac{x + 5}{2} - x \implies 2x - 1 = \frac{5(x + 5)}{2} - 5x \] En simplifiant, on trouve : \[ x = 3 \] 4. **Équation :** \( x^2 - 5 = 0 \) **Solution :** \[ x^2 - 5 = 0 \implies x^2 = 5 \implies x_{1} = -\sqrt{5}, \quad x_{2} = \sqrt{5} \] ### 2) Résoudre les inéquations suivantes : 1. **Inéquation :** \( 2x - 9 \geq 7 \) **Solution :** \[ 2x - 9 \geq 7 \implies 2x \geq 16 \implies x \geq 8 \] ### Résumé des solutions : - \( x = 2 \) - Pas de solution pour la deuxième équation. - \( x = 3 \) - \( x_{1} = -\sqrt{5}, \quad x_{2} = \sqrt{5} \) - \( x \geq 8 \) pour l'inéquation.