To find all the numbers [tex]\( n \)[/tex] that satisfy the given inequality "Three times two less than a number is greater than or equal to five times the number", we need to set up and solve the corresponding inequality step by step.
Let's start by interpreting the given statement mathematically:
1. Identify the components of the inequality:
- "Two less than a number" can be written as [tex]\( n - 2 \)[/tex].
- "Three times two less than a number" translates to [tex]\( 3(n - 2) \)[/tex].
- This is supposed to be "greater than or equal to five times the number", which we write as [tex]\( 5n \)[/tex].
2. Set up the inequality:
The verbal description gives us the following inequality:
[tex]\[
3(n - 2) \geq 5n
\][/tex]
3. Solve the inequality step by step:
- Distribute the 3 on the left-hand side:
[tex]\[
3n - 6 \geq 5n
\][/tex]
- Get all terms involving [tex]\( n \)[/tex] on one side of the inequality. To do this, subtract [tex]\( 5n \)[/tex] from both sides:
[tex]\[
3n - 5n - 6 \geq 0
\][/tex]
- Simplify the left-hand side:
[tex]\[
-2n - 6 \geq 0
\][/tex]
- Isolate [tex]\( n \)[/tex] by getting rid of the constant term. Start by adding 6 to both sides:
[tex]\[
-2n \geq 6
\][/tex]
- Divide both sides by -2. Note that dividing by a negative number reverses the direction of the inequality:
[tex]\[
n \leq -3
\][/tex]
4. Conclude the solution:
Thus, the inequality [tex]\( 3(n - 2) \geq 5n \)[/tex] is satisfied when [tex]\( n \leq -3 \)[/tex].
Therefore, all numbers [tex]\( n \)[/tex] that are less than or equal to -3 satisfy the given mathematical relationship.
