№1 а) 5х-3>4 б) 4(х-4)≤12 №2 {0,2(х-5)<2 {0,4(х+1)≥-1 №3 (х+2)^2(х-3)/х+1 ≤0 №4 |2х-4|<7 №5 а) 4х+2<7 б) 3(х+2)≥11 №6 {0,5(х-4)<4 {0,2(х+2)≥-2 №7 (х-4)^2(х+3)/(х-2) ≥0 №8 |3х+1|>4

Let's solve each of the inequalities step by step. ### №1 **a)** \( 5x - 3 > 4 \) 1. Add 3 to both sides: \[ 5x > 7 \] 2. Divide by 5: \[ x > \frac{7}{5} \] \[ x > 1.4 \] **b)** \( 4(x - 4) \leq 12 \) 1. Distribute 4: \[ 4x - 16 \leq 12 \] 2. Add 16 to both sides: \[ 4x \leq 28 \] 3. Divide by 4: \[ x \leq 7 \] ### Summary for №1: - **a)** \( x > 1.4 \) - **b)** \( x \leq 7 \) ### №2 **a)** \( 0.2(x - 5) < 2 \) 1. Multiply both sides by 10 (to eliminate the decimal): \[ 2(x - 5) < 20 \] 2. Distribute: \[ 2x - 10 < 20 \] 3. Add 10 to both sides: \[ 2x < 30 \] 4. Divide by 2: \[ x < 15 \] **b)** \( 0.4(x + 1) \geq -1 \) 1. Multiply both sides by 10: \[ 4(x + 1) \geq -10 \] 2. Distribute: \[ 4x + 4 \geq -10 \] 3. Subtract 4 from both sides: \[ 4x \geq -14 \] 4. Divide by 4: \[ x \geq -\frac{14}{4} \] \[ x \geq -3.5 \] ### Summary for №2: - **a)** \( x < 15 \) - **b)** \( x \geq -3.5 \) ### №3 \[ \frac{(x+2)^2(x-3)}{x+1} \leq 0 \] Identify critical points: - \( (x + 2)^2 = 0 \) ⇒ \( x = -2 \) (double root) - \( x - 3 = 0 \) ⇒ \( x = 3 \) - \( x + 1 = 0 \) ⇒ \( x = -1 \) (undefined) Test intervals: - For \( x < -2 \): Choose \( x = -3 \) → positive - For \( -2 < x < -1 \): Choose \( x = -1.5 \) → negative - For \( -1 < x < 3 \): Choose \( x = 0 \) → positive - For \( x > 3 \): Choose \( x = 4 \) → positive Summary of intervals: - Negative or zero on: \( -2 \leq x < -1 \) (include -2, not -1) ### Summary for №3: \[ x \in [-2, -1) \] ### №4 \[ |2x - 4| < 7 \] This results in two inequalities: 1. \( 2x - 4 < 7 \) \[ 2x < 11 \] \[ x < 5.5 \] 2. \( 2x - 4 > -7 \) \[ 2x > -3 \] \[ x > -\frac{3}{2} \] \[ x > -1.5 \] ### Summary for №4: \[ -1.5 < x < 5.5 \] ### №5 **a)** \( 4x + 2 < 7 \) 1. Subtract 2 from both sides: \[ 4x < 5 \] 2. Divide by 4: \[ x < \frac{5}{4} \] \[ x < 1.25 \] **b)** \( 3(x + 2) \geq 11 \) 1. Distribute: \[ 3x + 6 \geq 11 \] 2. Subtract 6 from both sides: \[ 3x \geq 5 \] 3. Divide by 3: \[ x \geq \frac{5}{3} \] \[ x \geq 1.6667 \] ### Summary for №5: - **a)** \( x < 1.25 \) - **b)** \( x \geq 1.6667 \) ### №6 **a)** \( 0.5(x - 4) < 4 \) 1. Multiply both sides by 2: \[ x - 4 < 8 \] 2. Add 4 to both sides: \[ x < 12 \] **b)** \( 0.2(x + 2) \geq -2 \) 1. Multiply both sides by 5: \[ x + 2 \geq -10 \] 2. Subtract 2 from both sides: \[ x \geq -12 \] ### Summary for №6: - **a)** \( x < 12 \) - **b)** \( x \geq -12 \) ### №7 \[ \frac{(x-4)^2(x+3)}{x-2} \geq 0 \] Critical points: - From \( (x-4)^2 = 0 \): \( x = 4 \) (double root) - From \( x + 3 = 0 \): \( x = -3 \) - From \( x - 2 = 0 \): \( x = 2 \) (undefined) Test intervals: - For \( x < -3 \): Choose \( x = -4 \) → positive - For \( -3 < x < 2 \): Choose \( x = 0 \) → negative - For \( 2 < x < 4 \): Choose \( x = 3 \) → positive - For \( x > 4 \): Choose \( x = 5 \) → positive ### Summary for №7: \[ x \in (-\infty, -3) \cup \left(2, 4\right] \] ### №8 \[ |3x + 1| > 4 \] This results in two inequalities: 1. \( 3x + 1 > 4 \) \[ 3x > 3 \] \[ x > 1 \] 2. \( 3x + 1 < -4 \) \[ 3x < -5 \] \[ x < -\frac{5}{3} \] ### Summary for №8: \[ x < -\frac{5}{3} \text{ or } x > 1 \] --- You can now summarize these results for your exam preparation! If you have any specific questions or need further clarifications, feel free to ask!