- Solve the first inequality \(-4x \leq 16\) by dividing by \(-4\), which flips the inequality sign to get \(x \geq -4\).
- Solve the second inequality \(4x + 5 < 25\) by subtracting 5 and dividing by 4 to get \(x < 5\).
- Combine the results to identify the valid range for \(x\), which is \(-4 \leq x < 5\).
- Match the derived inequality with the correct provided option, which is \(\boxed{-4 \leq x < 5}\).
Explanation
- Problem Analysis
We are given the compound inequality: \(-4x \leq 16\) AND \(4x + 5 < 25\). To solve this, we must determine the range of values for \(x\) that satisfy both inequalities simultaneously. We will solve each inequality individually and then combine the results.
- Solving the First Inequality
First, we solve the inequality \(-4x \leq 16\). To isolate \(x\), we divide both sides by \(-4\). It is a fundamental rule of algebra that when dividing or multiplying both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. Thus, dividing by \(-4\) gives:
\[x \geq \frac{16}{-4}\]
\[x \geq -4\]
- Solving the Second Inequality
Next, we solve the inequality \(4x + 5 < 25\). First, subtract 5 from both sides:
\[4x < 25 - 5\]
\[4x < 20\]
Now, divide by 4:
\[x < \frac{20}{4}\]
\[x < 5\]
- Combining Results and Identifying the Solution
To find the combined solution, we look for all values of \(x\) that are both greater than or equal to \(-4\) and strictly less than 5. We can write this as a compound inequality:
\[-4 \leq x < 5\]
Comparing this result to the provided options, we see it matches the third option: \(-4 \leq x < 5\).
Examples
Compound inequalities are essential in real-world scenarios where multiple constraints must be met simultaneously. For example, if you are designing a thermostat for a machine that operates safely only within a specific temperature range, say between 10 degrees Celsius and 30 degrees Celsius (inclusive of 10 but exclusive of 30), the operating range can be expressed as \(10 \leq T < 30\). Understanding how to solve and combine these inequalities ensures that systems function within their intended, safe parameters.