Let's solve the inequalities step by step:

a) $2x + 1 \geq 5 + x$

1. Subtract $x$ from both sides:
   $2x - x + 1 \geq 5$ Simplifying: $x + 1 \geq 5$
2. Subtract 1 from both sides: $x \geq 4$ This inequality is satisfied for $x \geq 4$.

Solution:

• The solution is $x \geq 4$, and on a number line, you will represent it by a closed circle on 4, shading to the right.

b) $2(x + 2) < -14 - x$

1. Distribute the 2 on the left side: $2x + 4 < -14 - x$
2. Add $x$ to both sides to get rid of $x$ on the right: $3x + 4 < -14$
3. Subtract 4 from both sides: $3x < -18$
4. Divide by 3: $x < -6$

Solution:

• The solution is $x < -6$, and on a number line, it’s represented by an open circle on -6, shading to the left.

c) $x - 6 \geq 4x + 3$

1. Subtract $x$ from both sides: $-6 \geq 3x + 3$
2. Subtract 3 from both sides: $-9 \geq 3x$
3. Divide by 3: $-3 \geq x$ or equivalently, $x \leq -3$.

Solution:

• The solution is $x \leq -3$, and on a number line, it’s represented by a closed circle on -3, shading to the left.

d) $-4(x - 5) \leq -3(2x - 7)$

1. Distribute both sides: $-4x + 20 \leq -6x + 21$
2. Add $6x$ to both sides: $2x + 20 \leq 21$
3. Subtract 20 from both sides: $2x \leq 1$
4. Divide by 2: $x \leq \frac{1}{2}$

Solution:

• The solution is $x \leq \frac{1}{2}$, and on a number line, it’s represented by a closed circle on $\frac{1}{2}$, shading to the left.

Would you like me to explain anything in more detail or provide a number line visualization?

Here are a few questions that can further expand your understanding:

1. How do inequalities behave differently when multiplying or dividing by negative numbers?
2. What is the significance of open and closed circles on the number line?
3. How can you graph a compound inequality?
4. How do you check if a solution to an inequality is correct?
5. What are the differences between strict inequalities ($<$, $>$) and non-strict inequalities ($\leq$, $\geq$)?

Tip: Always reverse the direction of the inequality when multiplying or dividing by a negative number.