# Solve and Graph An Inequality Example

Review how to solve and graph an inequality using an example that requires you to reverse the directions of the inequality sign.

Solve:

$-2x + 16 \geq 8$

Step 1: Subtract $$16$$ from both sides to isolate the term with the variable:

$-2x + 16 - 16 \geq 8 - 16$

This simplifies to:

$-2x \geq -8$

Step 2: Divide both sides by $$-2$$. Remember, when dividing or multiplying both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

$\frac{{-2x}}{{-2}} \leq \frac{{-8}}{{-2}}$

This simplifies to:

$x \leq 4$

So, the solution to the inequality is $$x \leq 4$$.

Explanation of the sign reversal:
When we multiplied both sides of the inequality by $$-2$$ in step 2, we needed to flip the inequality sign from $$\geq$$ to $$\leq$$. This is because when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign flips.

This principle can be understood intuitively. Consider the inequality $$-2x \geq 8$$. If we divide both sides by $$-2$$ without flipping the sign, we would get $$x \leq -4$$, which would imply that all numbers less than or equal to $$-4$$ satisfy the inequality. However, if you check the original inequality with a number greater than $$-4$$, say $$x = 0$$, you'll find that it satisfies the inequality ($$-2(0) + 16 \geq 8$$), which contradicts the original statement. Therefore, we must flip the sign to maintain the correct relationship between the numbers.