Let’s solve the inequalities step by step. I’ll provide solutions for all exercises in the image (7-20):

7. $x - 4 < -5$:

Add 4 to both sides: $x < -1$

8. $1 \leq s - 8$:

Add 8 to both sides: $9 \leq s \quad \text{or equivalently } s \geq 9$

9. $6 \geq m - 1$:

Add 1 to both sides: $7 \geq m \quad \text{or equivalently } m \leq 7$

10. $c - 12 > -4$:

Add 12 to both sides: $c > 8$

11. $r + 4 < 5$:

Subtract 4 from both sides: $r < 1$

12. $-8 \leq 8 + y$:

Subtract 8 from both sides: $-16 \leq y \quad \text{or equivalently } y \geq -16$

13. $9 + w > 7$:

Subtract 9 from both sides: $w > -2$

14. $15 \geq q + 3$:

Subtract 3 from both sides: $12 \geq q \quad \text{or equivalently } q \leq 12$

15. $h - (-2) \geq 10$:

Simplify $h - (-2)$ as $h + 2$, then subtract 2 from both sides: $h \geq 8$

16. $-6 > t - (-13)$:

Simplify $t - (-13)$ as $t + 13$, then subtract 13 from both sides: $-19 > t \quad \text{or equivalently } t < -19$

17. $j + 9 - 3 < 8$:

Simplify $j + 9 - 3$ to $j + 6$, then subtract 6 from both sides: $j < 2$

18. $1 - 12 + y \geq -5$:

Simplify $1 - 12 + y$ to $-11 + y$, then add 11 to both sides: $y \geq 6$

19. $10 \geq 3p - 2p - 7$:

Simplify $3p - 2p$ to $p$, so the inequality becomes $10 \geq p - 7$. Add 7 to both sides: $17 \geq p \quad \text{or equivalently } p \leq 17$

20. $18 - 5z + 6z \geq 3 + 6$:

Simplify $-5z + 6z$ to $z$, so the inequality becomes $18 + z \geq 9$. Subtract 18 from both sides: $z \geq -9$

Would you like me to explain any of these steps further or assist with graphing the solutions?

Relative Questions:

1. How can you graph the solution for a simple inequality?
2. What are the key differences in solving $\leq$ and $<$?
3. How can you interpret $t < -19$ on a number line?
4. How do multi-step inequalities differ from simple ones?
5. What are common errors to avoid in solving inequalities?

Tip:

Always reverse the inequality sign when multiplying or dividing by a negative number!