How to Solve Compound Inequalities: Step-by-Step Guide

Solving compound inequalities is a crucial skill in algebra that helps students understand the relationships between numbers. In this case, we are asked to solve two inequalities: $x + 1 \geq 5$ and $2x \geq 4$. Let’s break down the solution step-by-step.

### Step 1: Solve Each Inequality Separately

First, we need to tackle the inequality $x + 1 \geq 5$. To isolate $x$, we subtract 1 from both sides:

$$ x + 1 - 1 \geq 5 - 1 $$
$$ x \geq 4 $$

This tells us that $x$ must be greater than or equal to 4.

Next, we solve the second inequality, $2x \geq 4$. To do this, we divide both sides of the inequality by 2:

$$ \frac{2x}{2} \geq \frac{4}{2} $$
$$ x \geq 2 $$

This means that $x$ must also be greater than or equal to 2.

### Step 2: Find the Intersection of Solutions

Since the original question specifies that both inequalities must be satisfied (indicated by the word 'and'), we need to determine the overlap of the two solution sets:
- From the first inequality, we found $x \geq 4$.
- From the second inequality, we have $x \geq 2$.

To satisfy both conditions, we look for values of $x$ that satisfy both inequalities simultaneously. The more restrictive condition is $x \geq 4$, since any number greater than or equal to 4 will also be greater than or equal to 2.

### Conclusion

Therefore, the solution set to the compound inequality is:
$$ \{ x | x \geq 4 \} $$

### Real-World Application
Understanding how to solve inequalities is not just a mathematical exercise; it has real-world applications. For example, if a company wants to ensure that the number of products sold ($x$) meets a certain threshold to be profitable, knowing how to set and solve these inequalities helps in making informed decisions.

If you want to review the steps or have any questions, don't hesitate to check the equations section for more details on solving inequalities. Practicing with different types of inequalities will enhance your understanding and confidence in algebra!