To solve the compound inequality
[tex]\[2y - 4 > 0 \text{ or } 4y - 3 > -15,\][/tex]
we will tackle each inequality separately and then combine the results.
Step 1: Solve the inequality [tex]\(2y - 4 > 0\)[/tex].
1. Add 4 to both sides to isolate the term with [tex]\(y\)[/tex]:
[tex]\[2y - 4 + 4 > 0 + 4\][/tex]
[tex]\[2y > 4\][/tex]
2. Divide both sides by 2 to solve for [tex]\(y\)[/tex]:
[tex]\[\frac{2y}{2} > \frac{4}{2}\][/tex]
[tex]\[y > 2\][/tex]
So the solution for the first inequality is:
[tex]\[y > 2\][/tex]
Step 2: Solve the inequality [tex]\(4y - 3 > -15\)[/tex].
1. Add 3 to both sides to isolate the term with [tex]\(y\)[/tex]:
[tex]\[4y - 3 + 3 > -15 + 3\][/tex]
[tex]\[4y > -12\][/tex]
2. Divide both sides by 4 to solve for [tex]\(y\)[/tex]:
[tex]\[\frac{4y}{4} > \frac{-12}{4}\][/tex]
[tex]\[y > -3\][/tex]
So the solution for the second inequality is:
[tex]\[y > -3\][/tex]
Step 3: Combine the solutions of the inequalities.
The compound inequality uses "or," which means we take the union of the sets of solutions.
- The solution to [tex]\(2y - 4 > 0\)[/tex] is [tex]\(y > 2\)[/tex].
- The solution to [tex]\(4y - 3 > -15\)[/tex] is [tex]\(y > -3\)[/tex].
When taking the union of [tex]\(y > 2\)[/tex] and [tex]\(y > -3\)[/tex], the broader solution [tex]\(y > -3\)[/tex] encompasses the more restrictive solution [tex]\(y > 2\)[/tex].
Thus, the final solution to the compound inequality is:
[tex]\[y > -3\][/tex]
