Let's solve the inequality step by step:

Given inequality:

$2(x + 4) + 3 \geq 7 + 2x$

Step 1: Distribute the 2 on the left side.

$2x + 8 + 3 \geq 7 + 2x$ $2x + 11 \geq 7 + 2x$

Step 2: Subtract $2x$ from both sides.

$2x + 11 - 2x \geq 7 + 2x - 2x$ $11 \geq 7$

Step 3: Simplify.

$11 \geq 7$

This is always true, meaning the inequality holds for all real numbers.

Conclusion:

The solution set is all real numbers, so in interval notation, we write: $(-\infty, \infty)$

Would you like more details on any part of this explanation?

Related Questions:

1. How do we handle inequalities with fractions?
2. What are the rules for multiplying or dividing both sides of an inequality by a negative number?
3. How do we solve compound inequalities?
4. Can you explain how to graph solution sets of inequalities on a number line?
5. How do absolute value inequalities differ from regular inequalities?

Tip: When solving inequalities, if you multiply or divide both sides by a negative number, remember to reverse the inequality sign.