Final answer:
To solve the given system of inequalities, we isolated x in each individual inequality and then combined our solutions. The interval notation for the solution is (-\u221e, -1] ∪ (2, \u221e), representing all x-values less than or equal to -1 or greater than 2.
We also verified that our solution is reasonable by ensuring values within the intervals satisfy at least one of the inequalities.
Step-by-step explanation:
To solve the inequality 6x + 6 > 4x + 10 or -2 + 8x ≥ 15x + 5, we tackle each inequality separately. First, we simplify each inequality by eliminating terms where possible to isolate the variable x.
- For the first inequality: Subtract 4x from both sides to get 2x + 6 > 10. Then subtract 6 from both sides to get 2x > 4. Finally, divide both sides by 2 to solve for x, which gives us x > 2.
- For the second inequality: Subtract 8x from both sides to get -2 ≥ 7x + 5. Then subtract 5 from both sides to get -7 ≥ 7x. Finally, divide both sides by 7 to solve for x, which gives us x ≤ -1.
Now, we combine the solutions. The first inequality gives us values of x that are greater than 2. The second inequality gives us values of x that are less than or equal to -1. The solution to the system of inequalities, since it's an 'or' situation, includes all x-values that satisfy at least one of the inequalities.
Expressing the solution in interval notation, we get (-\u221e, -1] ∪ (2, \u221e), which means the solution includes all x-values less than or equal to -1 or greater than 2.
To check if our answer is reasonable, we see that the values within the solution intervals do indeed satisfy at least one of the original inequalities.