Solve the Linear Inequality: 4x + 7 > 19

Q: When do I need to flip the inequality sign?

Only when multiplying or dividing by a negative number! Since we divided by positive 4 in this problem, the > sign stays the same. If we divided by -4, then we'd flip it.

Q: How do I check if x > 3 is correct?

Pick any number greater than 3, like x = 5. Substitute: 4(5) + 7 = 27. Since 27 > 19, your answer is correct! You can also test x = 2: 4(2) + 7 = 15, and 15 is not > 19.

Q: What does the solution x > 3 actually mean?

It means any number greater than 3 makes the inequality true! This includes 3.1, 4, 10, 100, etc. The number 3 itself does not work because we need strictly greater than.

Q: Can I solve this by guessing and checking?

You could, but it's not efficient! Algebraic methods like the one shown give you the exact answer quickly. Guessing only helps verify your solution afterward.

Linear Inequalities with Basic Operations

Solve the inequality:

$4x + 7 > 19$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the inequality:

$4x + 7 > 19$

Step-by-step solution

To solve the inequality $4x + 7 > 19$ , follow these steps:

1. Subtract 7 from both sides to isolate the term with $x$ on one side.

$4x + 7 - 7 > 19 - 7$

2. This simplifies to:

$4x > 12$

3. Next, divide each side by 4 to solve for $x$ .

$x > \frac{12}{4}$

4. This simplifies further:

$x > 3$

Therefore, the solution is $x > 3$ .

Final Answer

$x > 3$

Key Points to Remember

Essential concepts to master this topic

Rule: Perform same operations on both sides to maintain inequality
Technique: Subtract 7 first: 4x + 7 - 7 > 19 - 7
Check: Test x = 4: 4(4) + 7 = 23 > 19 ✓

Common Mistakes

Avoid these frequent errors

Flipping inequality sign when dividing by positive number
Don't flip the > sign when dividing 4x > 12 by 4 = wrong direction! The inequality only flips when multiplying or dividing by negative numbers. Always keep the same direction when dividing by positive numbers.

Practice Quiz

Test your knowledge with interactive questions

Solve the following inequality:

$3x+4 \leq 10$

FAQ

Everything you need to know about this question

When do I need to flip the inequality sign?

Only when multiplying or dividing by a negative number! Since we divided by positive 4 in this problem, the > sign stays the same. If we divided by -4, then we'd flip it.

How is solving inequalities different from equations?

The steps are almost identical! The main difference is that you get a range of solutions (like x > 3) instead of just one number. Plus, watch out for that sign-flipping rule!

How do I check if x > 3 is correct?

Pick any number greater than 3, like x = 5. Substitute: 4(5) + 7 = 27. Since 27 > 19, your answer is correct! You can also test x = 2: 4(2) + 7 = 15, and 15 is not > 19.

What does the solution x > 3 actually mean?

It means any number greater than 3 makes the inequality true! This includes 3.1, 4, 10, 100, etc. The number 3 itself does not work because we need strictly greater than.

Can I solve this by guessing and checking?

You could, but it's not efficient! Algebraic methods like the one shown give you the exact answer quickly. Guessing only helps verify your solution afterward.

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