To solve the given problem of finding all numbers that satisfy the condition "Three times two less than a number is greater than or equal to five times the number," we need to translate this verbal statement into a mathematical inequality and solve it step-by-step.
1. Translate the statement into a mathematical inequality:
Given:
- A number is represented by [tex]\( n \)[/tex].
- "Two less than a number" is represented by [tex]\( n - 2 \)[/tex].
- "Three times two less than a number" translates to [tex]\( 3(n - 2) \)[/tex].
- This expression needs to be greater than or equal to five times the number ([tex]\( 5n \)[/tex]).
So, the inequality that represents this relationship is:
[tex]\[
3(n - 2) \geq 5n
\][/tex]
2. Solve the inequality step-by-step:
Step 1: Distribute the 3 on the left side of the inequality:
[tex]\[
3(n - 2) = 3n - 6
\][/tex]
Therefore, the inequality becomes:
[tex]\[
3n - 6 \geq 5n
\][/tex]
Step 2: Move all terms involving [tex]\( n \)[/tex] to one side by subtracting [tex]\( 3n \)[/tex] from both sides:
[tex]\[
3n - 6 - 3n \geq 5n - 3n
\][/tex]
Simplifying the terms:
[tex]\[
-6 \geq 2n
\][/tex]
Step 3: Divide both sides by 2 to isolate [tex]\( n \)[/tex]:
[tex]\[
\frac{-6}{2} \geq \frac{2n}{2}
\][/tex]
[tex]\[
-3 \geq n
\][/tex]
This simplifies to:
[tex]\[
n \leq -3
\][/tex]
Thus, the solution to the inequality is [tex]\( n \leq -3 \)[/tex]. In other words, any number less than or equal to [tex]\(-3\)[/tex] will satisfy the given condition that three times two less than the number is greater than or equal to five times the number.
