Question

what is the missing step in solving the inequality \\(4(x - 3) + 4 \le 10 + 6x\\)?

1. the distributive property: \\(4x - 12 + 4 \le 10 + 6x\\)
2. combine like terms: \\(4x - 8 \le 10 + 6x\\)
3. the addition property of inequality: \\(4x \le 18 + 6x\\)
4. the subtraction property of inequality: \\(-2x \le 18\\)
5. the division property of inequality: _______

\\(x \le -9\\)
\\(x \ge -9\\)
\\(x \le -\frac{1}{9}\\)
\\(x \ge -\frac{1}{9}\\)

Explanation:

Identify the current state of the inequality

The problem asks for the missing step 5 in solving the inequality.
From step 4, we have the inequality:

$$ -2x \le 18 $$

Apply the division property of inequality

To isolate \(x\), we need to divide both sides of the inequality by \(-2\).
According to the properties of inequalities, when dividing or multiplying both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed.

Calculate the final simplified inequality

Dividing both sides by \(-2\):

$$ \frac{-2x}{-2} \ge \frac{18}{-2} $$

This simplifies to:

$$ x \ge -9 $$

Match with the given multiple-choice options

Let's compare our result \(x \ge -9\) with the options:

• Option 1: \(x \le -9\)
• Option 2: \(x \ge -9\)
• Option 3: \(x \le -\frac{1}{9}\)
• Option 4: \(x \ge -\frac{1}{9}\)

The correct option is the second one: \(x \ge -9\).

Answer:

• \(x \le -9\)
• \(x \ge -9\) (Correct answer)
• \(x \le -\frac{1}{9}\)
• \(x \ge -\frac{1}{9}\)