Sure, let's solve the inequality [tex]\(\frac{-6t}{t-2} < \frac{-30}{t-2}\)[/tex] using an algebraic approach, step by step.
1. Given Inequality:
[tex]\[
\frac{-6t}{t-2} < \frac{-30}{t-2}
\][/tex]
2. Identify and Cancel Common Denominators:
Notice that both sides of the inequality have the denominator [tex]\((t-2)\)[/tex]. We can multiply both sides by [tex]\((t-2)\)[/tex] to eliminate the denominators. However, we need to be careful about the sign of [tex]\((t-2)\)[/tex]. We will consider two cases: [tex]\(t > 2\)[/tex] and [tex]\(t < 2\)[/tex].
3. Case 1: [tex]\(t > 2\)[/tex]
- When [tex]\(t > 2\)[/tex], [tex]\((t-2) > 0\)[/tex]. So multiplying both sides by the positive term [tex]\((t-2)\)[/tex] does not change the direction of the inequality:
[tex]\[
-6t < -30
\][/tex]
- To solve [tex]\(-6t < -30\)[/tex], divide both sides by [tex]\(-6\)[/tex] and reverse the inequality (since dividing by a negative number reverses the inequality sign):
[tex]\[
t > 5
\][/tex]
- Therefore, for [tex]\(t > 2\)[/tex], the inequality implies [tex]\(t > 5\)[/tex].
4. Case 2: [tex]\(t < 2\)[/tex]
- When [tex]\(t < 2\)[/tex], [tex]\((t-2) < 0\)[/tex]. So multiplying both sides by the negative term [tex]\((t-2)\)[/tex] reverses the direction of the inequality:
[tex]\[
-6t > -30
\][/tex]
- To solve [tex]\(-6t > -30\)[/tex], divide both sides by [tex]\(-6\)[/tex] and reverse the inequality (since dividing by a negative number reverses the inequality sign):
[tex]\[
t < 5
\][/tex]
- Therefore, for [tex]\(t < 2\)[/tex], the inequality is satisfied for [tex]\(t < 5\)[/tex].
5. Combine the Two Cases:
- From the first case, for [tex]\(t > 2\)[/tex], [tex]\(t\)[/tex] must be greater than 5.
- From the second case, for [tex]\(t < 2\)[/tex], [tex]\(t\)[/tex] can be any value less than 2 (except exactly 2 because it would make the denominator zero).
6. Final Solution:
[tex]\[
t < 2 \text{ or } t > 5
\][/tex]
Therefore, the solution to the inequality [tex]\(\frac{-6t}{t-2} < \frac{-30}{t-2}\)[/tex] is [tex]\(t < 2\)[/tex] or [tex]\(t > 5\)[/tex].
