Here is the solution for the exercise based on the image provided:

Exercise 2: Solve the following inequalities and equations

1. Solve the inequalities:

(a) $3x + 2 \leq 3$

• Subtract 2 from both sides: $3x \leq 1$
• Divide by 3: $x \leq \frac{1}{3}$

Solution: $x \in \left(-\infty, \frac{1}{3} \right]$

(b) $2x - 3 \geq 2$

• Add 3 to both sides: $2x \geq 5$
• Divide by 2: $x \geq \frac{5}{2}$

Solution: $x \in \left[\frac{5}{2}, \infty \right)$

(c) $x^2 - 6x + 5 \geq 0$

• Solve the quadratic equation $x^2 - 6x + 5 = 0$ by factoring: $(x - 1)(x - 5) = 0$
• Roots: $x = 1$ and $x = 5$
• Test intervals $(-\infty, 1)$, $(1, 5)$, $(5, \infty)$ using a sign table:
  • For $x \in (-\infty, 1): (x - 1)(x - 5) > 0$
  • For $x \in (1, 5): (x - 1)(x - 5) < 0$
  • For $x \in (5, \infty): (x - 1)(x - 5) > 0$

Solution: $x \in (-\infty, 1] \cup [5, \infty)$

(d) $x^2 - 6x + 5 < 0$

• Use the roots from part (c): $x = 1$ and $x = 5$
• From the sign table:
  • For $x \in (1, 5): (x - 1)(x - 5) < 0$

Solution: $x \in (1, 5)$

2. Solve in $\mathbb{R}$:

System of equations: