Answer :
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.
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.
