Let's carefully translate the given word problem into a mathematical inequality and solve it step by step.
The problem statement says: "Three times a number plus four is no more than four times the number minus one."
Let's denote the unknown number by [tex]\( z \)[/tex].
We translate the statement into an inequality as follows:
[tex]\[ 3z + 4 \leq 4z - 1 \][/tex]
Next, we aim to isolate [tex]\( z \)[/tex] on one side of the inequality. We will start by subtracting [tex]\( 3z \)[/tex] from both sides:
[tex]\[ 3z + 4 - 3z \leq 4z - 1 - 3z \][/tex]
[tex]\[ 4 \leq z - 1 \][/tex]
Then, we add 1 to both sides of the inequality to solve for [tex]\( z \)[/tex]:
[tex]\[ 4 + 1 \leq z - 1 + 1 \][/tex]
[tex]\[ 5 \leq z \][/tex]
Thus, we can write this as:
[tex]\[ z \geq 5 \][/tex]
So, the correct interpretation of the solution to this problem is that the number [tex]\( z \)[/tex] must be at least 5.
Now, let's evaluate which of the given statements are true:
1. [tex]\( z > 5 \)[/tex] — This statement is not true because [tex]\( z \)[/tex] can be exactly 5.
2. [tex]\( z \geq 5 \)[/tex] — This statement is true.
3. 255 — This statement appears to be incorrect and irrelevant to the context.
4. "The number is at least 5." — This statement is true.
5. "The number is at most 5." — This statement is not true because [tex]\( z \)[/tex] can be greater than 5.
Therefore, the true statements are:
- [tex]\( z \geq 5 \)[/tex]
- "The number is at least 5."
