EXERCISE 1 Solve for \( x \) in each of the following equations and check each solution. 1. \( 3 x-4=8 \) 2. \( -5(x-2)+4 x=x(3-x)-4(x-2)+x^{2}+2 \) 3. \( 3(2 x-1)+4=3-5(1-x) \) 4. \( 2(x-3)+5=7-2(1-x) \) 5. \( x-(x-2)-2(x-3)=0 \) 6. \( x(2-3 x)+(x+2)(3 x-2)=0 \) 7. \( -2+(x-1)(x+2)=x(x-3)+4(x-1) \) 8. \( (x-3)(x+2)=x(x+1)-3(x+2)+3 \) 9. \( (x+3)(x+4)=4-x(5-x) \) 10. \( 2(x-3)-x(x-2)=(3-x)(x-2)-x \)

Solve the equation by following steps: - step0: Solve for \(x\): \(x-\left(x-2\right)-2\left(x-3\right)=0\) - step1: Subtract the terms: \(2-2\left(x-3\right)=0\) - step2: Calculate: \(8-2x=0\) - step3: Move the constant to the right side: \(-2x=0-8\) - step4: Remove 0: \(-2x=-8\) - step5: Change the signs: \(2x=8\) - step6: Divide both sides: \(\frac{2x}{2}=\frac{8}{2}\) - step7: Divide the numbers: \(x=4\) Solve the equation \( -5(x-2)+4x=x(3-x)-4(x-2)+x^{2}+2 \). Solve the equation by following steps: - step0: Solve for \(x\): \(-5\left(x-2\right)+4x=x\left(3-x\right)-4\left(x-2\right)+x^{2}+2\) - step1: Add the terms: \(-5\left(x-2\right)+4x=3x-4\left(x-2\right)+2\) - step2: Move the expression to the left side: \(-5\left(x-2\right)+4x-\left(3x-4\left(x-2\right)+2\right)=0\) - step3: Calculate: \(-5\left(x-2\right)+x+4\left(x-2\right)-2=0\) - step4: Calculate: \(0=0\) - step5: The statement is true: \(x \in \mathbb{R}\) Solve the equation \( 2(x-3)-x(x-2)=(3-x)(x-2)-x \). Solve the equation by following steps: - step0: Solve for \(x\): \(2\left(x-3\right)-x\left(x-2\right)=\left(3-x\right)\left(x-2\right)-x\) - step1: Move the expression to the left side: \(2\left(x-3\right)-x\left(x-2\right)-\left(\left(3-x\right)\left(x-2\right)-x\right)=0\) - step2: Subtract the terms: \(0=0\) - step3: The statement is true: \(x \in \mathbb{R}\) Solve the equation \( x(2-3x)+(x+2)(3x-2)=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(x\left(2-3x\right)+\left(x+2\right)\left(3x-2\right)=0\) - step1: Add the terms: \(2\left(3x-2\right)=0\) - step2: Rewrite the expression: \(3x-2=0\) - step3: Move the constant to the right side: \(3x=0+2\) - step4: Remove 0: \(3x=2\) - step5: Divide both sides: \(\frac{3x}{3}=\frac{2}{3}\) - step6: Divide the numbers: \(x=\frac{2}{3}\) Solve the equation \( (x+3)(x+4)=4-x(5-x) \). Solve the equation by following steps: - step0: Solve for \(x\): \(\left(x+3\right)\left(x+4\right)=4-x\left(5-x\right)\) - step1: Calculate: \(x^{2}+7x+12=4-x\left(5-x\right)\) - step2: Calculate: \(x^{2}+7x+12=4-5x+x^{2}\) - step3: Move the expression to the left side: \(x^{2}+7x+12-\left(4-5x+x^{2}\right)=0\) - step4: Calculate: \(12x+8=0\) - step5: Move the constant to the right side: \(12x=0-8\) - step6: Remove 0: \(12x=-8\) - step7: Divide both sides: \(\frac{12x}{12}=\frac{-8}{12}\) - step8: Divide the numbers: \(x=-\frac{2}{3}\) Solve the equation \( -2+(x-1)(x+2)=x(x-3)+4(x-1) \). Solve the equation by following steps: - step0: Solve for \(x\): \(-2+\left(x-1\right)\left(x+2\right)=x\left(x-3\right)+4\left(x-1\right)\) - step1: Move the expression to the left side: \(-2+\left(x-1\right)\left(x+2\right)-\left(x\left(x-3\right)+4\left(x-1\right)\right)=0\) - step2: Remove the parentheses: \(-2+\left(x-1\right)\left(x+2\right)-x\left(x-3\right)-4\left(x-1\right)=0\) - step3: Calculate: \(0=0\) - step4: The statement is true: \(x \in \mathbb{R}\) Solve the equation \( 3x-4=8 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3x-4=8\) - step1: Move the constant to the right side: \(3x=8+4\) - step2: Add the numbers: \(3x=12\) - step3: Divide both sides: \(\frac{3x}{3}=\frac{12}{3}\) - step4: Divide the numbers: \(x=4\) Solve the equation \( 3(2x-1)+4=3-5(1-x) \). Solve the equation by following steps: - step0: Solve for \(x\): \(3\left(2x-1\right)+4=3-5\left(1-x\right)\) - step1: Move the expression to the left side: \(3\left(2x-1\right)+4-\left(3-5\left(1-x\right)\right)=0\) - step2: Calculate: \(3\left(2x-1\right)+1+5\left(1-x\right)=0\) - step3: Calculate: \(x+3=0\) - step4: Move the constant to the right side: \(x=0-3\) - step5: Remove 0: \(x=-3\) Solve the equation \( 2(x-3)+5=7-2(1-x) \). Solve the equation by following steps: - step0: Solve for \(x\): \(2\left(x-3\right)+5=7-2\left(1-x\right)\) - step1: Move the expression to the left side: \(2\left(x-3\right)+5-\left(7-2\left(1-x\right)\right)=0\) - step2: Calculate: \(-6=0\) - step3: The statement is false: \(x \in \varnothing \) Solve the equation \( (x-3)(x+2)=x(x+1)-3(x+2)+3 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\left(x-3\right)\left(x+2\right)=x\left(x+1\right)-3\left(x+2\right)+3\) - step1: Calculate: \(x^{2}-x-6=x\left(x+1\right)-3\left(x+2\right)+3\) - step2: Calculate: \(x^{2}-x-6=x^{2}-2x-3\) - step3: Move the expression to the left side: \(x^{2}-x-6-\left(x^{2}-2x-3\right)=0\) - step4: Calculate: \(x-3=0\) - step5: Move the constant to the right side: \(x=0+3\) - step6: Remove 0: \(x=3\) Let's solve each equation step by step and check the solutions. ### 1. Solve \( 3x - 4 = 8 \) **Step 1:** Add 4 to both sides: \[ 3x = 12 \] **Step 2:** Divide by 3: \[ x = 4 \] **Check:** Substitute \( x = 4 \) back into the original equation: \[ 3(4) - 4 = 12 - 4 = 8 \quad \text{(True)} \] **Solution:** \( x = 4 \) --- ### 2. Solve \( -5(x-2) + 4x = x(3-x) - 4(x-2) + x^2 + 2 \) **Step 1:** Expand both sides: \[ -5x + 10 + 4x = 3x - x^2 - 4x + 8 + x^2 + 2 \] **Step 2:** Combine like terms: \[ -x + 10 = -x + 10 \] This is true for all \( x \). **Check:** All real numbers satisfy the equation. **Solution:** \( x \in \mathbb{R} \) (All real solutions) --- ### 3. Solve \( 3(2x - 1) + 4 = 3 - 5(1 - x) \) **Step 1:** Expand both sides: \[ 6x - 3 + 4 = 3 - 5 + 5x \] **Step 2:** Simplify: \[ 6x + 1 = -2 + 5x \] **Step 3:** Rearrange: \[ 6x - 5x = -2 - 1 \] \[ x = -3 \] **Check:** Substitute \( x = -3 \): \[ 3(2(-3) - 1) + 4 = 3 - 5(1 - (-3)) \] \[ 3(-6 - 1) + 4 = 3 - 5(4) \] \[ -21 + 4 = 3 - 20 \] \[ -17 = -17 \quad \text{(True)} \] **Solution:** \( x = -3 \) --- ### 4. Solve \( 2(x - 3) + 5 = 7 - 2(1 - x) \) **Step 1:** Expand both sides: \[ 2x - 6 + 5 = 7 - 2 + 2x \] **Step 2:** Simplify: \[ 2x - 1 = 5 + 2x \] **Step 3:** Rearrange: \[ -1 = 5 \quad \text{(False)} \] **Check:** No solutions exist. **Solution:** \( x \in \varnothing \) (No solution) --- ### 5. Solve \( x - (x - 2) - 2(x - 3) = 0 \) **Step 1:** Expand: \[ x - x + 2 - 2x + 6 = 0 \] **Step 2:** Combine like terms: \[ -2x + 8 = 0 \] **Step 3:** Solve for \( x \): \[ 2x = 8 \quad \Rightarrow \quad x = 4 \] **Check:** Substitute \( x = 4 \): \[ 4 - (4 - 2) - 2(4 - 3) = 0 \] \[ 4 - 2 - 2 = 0 \quad \text{(True)} \] **Solution:** \( x = 4 \) --- ### 6. Solve \( x(2 - 3x) + (x + 2)(3x - 2) = 0 \) **Step 1:** Expand: \[ 2x - 3x^2 + 3x^2 + 6x - 2 = 0 \] **Step 2:** Combine like terms: \[ 8x - 2 = 0 \] **Step 3:** Solve for \( x \): \[ 8x = 2 \quad \Rightarrow \quad x = \frac{1}{4} \] **Check:** Substitute \( x = \frac{1}{4} \): \[ \frac{1}{4}(2 - 3(\frac{1}{4})) + (\frac{1}{4} + 2)(3(\frac{1}{4}) - 2) = 0 \] \[ \frac{1}{4}(2 - \frac{3}{4}) + (\frac{1}{4} + 2)(\frac{3}{4} - 2) = 0 \] \[ \frac{1}{4}(\frac{5}{4}) + (\frac{9}{4})(-\frac{5}{4}) = 0 \] \[ \frac{5}{16} - \frac{45}{16} = 0 \quad \text{(True)} \] **Solution:** \( x = \frac{1}{4} \) --- ### 7. Solve \( -2 + (x - 1)(x + 2) = x(x - 3) + 4(x - 1) \) **Step 1:** Expand: \[ -2 + (x^2 + x - 2) = x^2 - 3x + 4x - 4 \] **Step 2:** Combine like terms: \[ x^2 + x - 2 - 2 = x^2 + x - 4 \] **Step 3:** Rearrange: \[ -4 = -4 \quad \text{(True)} \] **Check:** All real numbers satisfy the equation. **Solution:** \( x \in \mathbb{R} \) (All real solutions) --- ### 8. Solve \( (x - 3)(x + 2) = x(x + 1) - 3(x + 2) + 3 \) **Step 1:** Expand: \[ x^2 - x - 6 = x^2 + x - 3x - 6 + 3 \] **Step 2:** Combine like terms: \[ x^2 - x - 6 = x^2 - 2x - 3 \] **Step 3:** Rearrange: \[ -x + 2x - 6 + 3 = 0 \] \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] **Check:** Substitute \( x = 3 \): \[ (3 - 3)(3 + 2) = 3(3 + 1) - 3(