№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:

\[ x < -\frac{5}{3} \text{ or } x > 1 \]

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You can now summarize these results for your exam preparation! If you have any specific questions or need further clarifications, feel free to ask!