Question

if three times a number is at least four more than eight times the number, which of the following is true?

the number is at least negative four-fifths.
the number is at most four-fifths.
the number is at least four-fifths.
the number is at most negative four-fifths.

Explanation:

Translate the word problem into an inequality

Let \(x\) represent the unknown number.

• "Three times a number" is represented as \(3x\).
• "At least" means greater than or equal to (\(\ge\)).
• "Four more than eight times the number" is represented as \(8x + 4\).

Putting it all together, we get the inequality:

$$3x \ge 8x + 4$$

Solve the inequality for x

Subtract \(8x\) from both sides of the inequality:

$$3x - 8x \ge 4$$

$$-5x \ge 4$$

Divide both sides by \(-5\). Remember that dividing or multiplying an inequality by a negative number reverses the inequality sign:

$$x \le -\frac{4}{5}$$

Interpret the solution in words

The inequality \(x \le -\frac{4}{5}\) means that the number \(x\) is less than or equal to negative four-fifths.
In other words, the number is at most negative four-fifths.

Answer:

• The number is at least negative four-fifths.
• The number is at most four-fifths.
• The number is at least four-fifths.
• The number is at most negative four-fifths. (Correct answer)