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.

 

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