Sure, let's solve the given inequalities step-by-step.
### First Inequality: [tex]\(\frac{a}{-2} < -1\)[/tex]
1. Start with the given inequality:
[tex]\[\frac{a}{-2} < -1\][/tex]
2. To isolate [tex]\(a\)[/tex], multiply both sides of the inequality by [tex]\(-2\)[/tex]. Remember, when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign:
[tex]\[a > (-1) \times (-2)\][/tex]
3. Simplify the right side:
[tex]\[a > 2\][/tex]
So, the first inequality simplifies to:
[tex]\[a > 2\][/tex]
### Second Inequality: [tex]\(-4a + 3 \geq 23\)[/tex]
1. Start with the given inequality:
[tex]\[-4a + 3 \geq 23\][/tex]
2. Subtract 3 from both sides to isolate the term with [tex]\(a\)[/tex]:
[tex]\[-4a \geq 23 - 3\][/tex]
3. Simplify the right side:
[tex]\[-4a \geq 20\][/tex]
4. Now, divide both sides of the inequality by [tex]\(-4\)[/tex]. Again, remember to reverse the inequality sign because we are dividing by a negative number:
[tex]\[a \leq \frac{20}{-4}\][/tex]
5. Simplify the right side:
[tex]\[a \leq -5\][/tex]
So, the second inequality simplifies to:
[tex]\[a \leq -5\][/tex]
### Combined Solution
The original compound inequality is connected by an "or" operator, meaning that [tex]\(a\)[/tex] can satisfy either one of the inequalities or both.
Thus, the solution set is:
[tex]\[a > 2 \text{ or } a \leq -5\][/tex]
In interval notation, this can be written as:
[tex]\[(2, \infty) \cup (-\infty, -5]\][/tex]
Therefore, the complete solution to the given inequality [tex]\(\frac{a}{-2} < -1\)[/tex] or [tex]\(-4a + 3 \geq 23\)[/tex] is:
[tex]\[a > 2 \text{ or } a \leq -5\][/tex]
