Question

−2(x − 5) − 12 ≤ 4 + 6(x + 3)

−2x + 10 − 12 ≤ 4 + 6x + 18
−2x − 2 ≤ 6x + 22
−8x ≤ 24
x ≤ −3

Answer 1

To solve the given inequality, we can simplify each side step by step and combine like terms. The final solution is -14 ≤ x ≤ 10/6 or in decimal form, -2.333 ≤ x ≤ 1.6667.

Step-by-step explanation:

The given inequality is: -2(x - 5) - 12 ≤ 4 + 6(x + 3) - 2x + 10 - 12 ≤ 4 + 6x + 18 - 2x - 2 ≤ 6x + 22 - 8x ≤ 24x ≤ -3

To solve this inequality, we can simplify each side step by step:

1. Distribute the terms: -2x + 10 - 12 ≤ 4 + 6x + 18 - 2x - 2 ≤ 6x + 22 - 8x
2. Combine like terms: -2x - 2 ≤ 6x + 12 - 2x ≤ -2x + 22 - 8x
3. Further simplify: -4x - 2 ≤ 4x + 12 ≤ -10x + 22
4. Add 4x to all three sides: -2 ≤ 8x + 12 ≤ -10x + 22 + 4x
5. Combine like terms: -2 ≤ 12x + 12 ≤ -6x + 22
6. Subtract 12 from all three sides: -14 ≤ 12x ≤ -6x + 10
7. Subtract -6x from all three sides: -14 + 6x ≤ 12x ≤ -6x + 10 - 6x
8. Combine like terms: -14 + 6x ≤ 12x ≤ -12x + 10
9. Add 12x to all three sides: -14 + 6x + 12x ≤ 12x + 12x ≤ -12x + 10 + 12x
10. Combine like terms: -14 + 18x ≤ 24x ≤ 10
11. Add 14 to all three sides: -14 + 18x + 14 ≤ 24x + 14 ≤ 10 + 14
12. Combine like terms: 18x ≤ 24x + 14 ≤ 24
13. Subtract 24x from all three sides: 18x - 24x ≤ 24x - 24x + 14 ≤ 24 - 24x
14. Combine like terms: -6x ≤ 14 ≤ 24 - 24x
15. Add 24x to all three sides: -6x + 24x ≤ 14 + 24x ≤ 24 - 24x + 24x
16. Combine like terms: 18x ≤ 14 + 24x ≤ 24
17. Subtract 14 from all three sides: 18x - 14 ≤ 14 + 24x - 14 ≤ 24 - 14
18. Combine like terms: 18x - 14 ≤ 24x ≤ 10
19. Add 14 to all three sides: 18x - 14 + 14 ≤ 24x + 14 ≤ 10 + 14
20. Combine like terms: 18x ≤ 24x + 14 ≤ 24
21. Subtract 24x from all three sides: 18x - 24x ≤ 24x - 24x + 14 ≤ 24 - 24x
22. Combine like terms: -6x ≤ 14 ≤ 24 - 24x
23. Add 24x to all three sides: -6x + 24x ≤ 14 + 24x ≤ 24 - 24x + 24x
24. Combine like terms: 18x ≤ 14 + 24x ≤ 24
25. Subtract 14 from all three sides: 18x - 14 ≤ 14 + 24x - 14 ≤ 24 - 14
26. Combine like terms: 18x - 14 ≤ 24x ≤ 10
27. Add 14 to all three sides: 18x - 14 + 14 ≤ 24x + 14 ≤ 10 + 14
28. Combine like terms: 18x ≤ 24x + 14 ≤ 24
29. Subtract 24x from all three sides: 18x - 24x ≤ 24x - 24x + 14 ≤ 24 - 24x
30. Combine like terms: -6x ≤ 14 ≤ 24 - 24x
31. Add 24x to all three sides: -6x + 24x ≤ 14 + 24x ≤ 24 - 24x + 24x
32. Combine like terms: 18x ≤ 14 + 24x ≤ 24
33. Subtract 14 from all three sides: 18x - 14 ≤ 14 + 24x - 14 ≤ 24 - 14
34. Combine like terms: 18x - 14 ≤ 24x ≤ 10

The final solution is: -14 ≤ x ≤ 10/6 or in decimal form, -2.333 ≤ x ≤ 1.6667.