Solve The Inequality: 5x - 2 > X + 10

Solve the Inequality: 5x - 2 > x + 10

Welcome to our math discussion! Today, we're going to tackle a common type of problem: solving linear inequalities. Specifically, we'll be working through the inequality 5x - 2 > x + 10. Inequalities are similar to equations, but instead of looking for a single value that makes the statement true, we're looking for a range of values. Let's break down the process step-by-step to find the solution.

Understanding Inequalities

Before we dive into the specific problem, it's helpful to remember what inequalities represent. An inequality is a mathematical statement that compares two expressions using symbols like ">" (greater than), "<" (less than), "≥" (greater than or equal to), and "≤" (less than or equal to). When we solve an inequality, we're aiming to isolate the variable (in this case, 'x') on one side of the inequality sign, just like we would with an equation. However, there's one crucial rule to keep in mind: if you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is a key difference from solving equations.

Our goal with 5x - 2 > x + 10 is to find all the values of 'x' that make the left side (5x - 2) larger than the right side (x + 10). Think of it like a scale. We want to adjust the scale so that the weight on the left is always heavier than the weight on the right. We can perform the same operations on both sides of the inequality as we would with an equation (adding, subtracting, multiplying, or dividing) to help us achieve this balance and isolate 'x'. However, we must always be mindful of that rule about multiplying or dividing by negative numbers.

Step-by-Step Solution

Let's start solving 5x - 2 > x + 10. Our first objective is to gather all the 'x' terms on one side of the inequality and all the constant terms on the other. It's generally easier to work with positive coefficients for 'x', so let's aim to move the 'x' terms to the left side.

1. Subtract 'x' from both sides: To get all the 'x' terms on the left, we subtract 'x' from both sides of the inequality. This doesn't change the direction of the inequality sign.

   5x - 2 - x > x + 10 - x

   This simplifies to:

   4x - 2 > 10

2. Add 2 to both sides: Now, we want to move the constant terms to the right side. We do this by adding 2 to both sides of the inequality. Again, this operation does not change the direction of the inequality sign.

   4x - 2 + 2 > 10 + 2

   This simplifies to:

   4x > 12

3. Divide both sides by 4: Finally, to isolate 'x', we divide both sides by 4. Since 4 is a positive number, the direction of the inequality sign remains unchanged.

   (4x) / 4 > 12 / 4

   This gives us our final solution:

   x > 3

So, the solution to the inequality 5x - 2 > x + 10 is x > 3. This means any number greater than 3 will make the original inequality true. For example, if x = 4, then 5(4) - 2 = 20 - 2 = 18, and 4 + 10 = 14. Since 18 > 14, the inequality holds true. If x = 3, then 5(3) - 2 = 15 - 2 = 13, and 3 + 10 = 13. Since 13 is not greater than 13, x = 3 is not part of the solution. If x = 2, then 5(2) - 2 = 10 - 2 = 8, and 2 + 10 = 12. Since 8 is not greater than 12, x = 2 is also not part of the solution.

Verifying the Solution

It's always a good practice to check your work. We found that the solution is x > 3. Let's pick a value greater than 3, say x = 5, and substitute it back into the original inequality:

5x - 2 > x + 10

Substitute x = 5:

5(5) - 2 > 5 + 10

25 - 2 > 15

23 > 15

This statement is true, which supports our solution.

Now, let's pick a value that is not greater than 3, say x = 3 (the boundary) or x = 0 (less than 3), to ensure it doesn't satisfy the inequality.

If x = 3:

5(3) - 2 > 3 + 10

15 - 2 > 13

13 > 13

This statement is false, as 13 is equal to 13, not greater than 13.

If x = 0:

5(0) - 2 > 0 + 10

0 - 2 > 10

-2 > 10

This statement is also false.

These checks confirm that our solution, x > 3, is correct.

Understanding the Options

We were given several options for the solution:

A. x > -3 B. x > 0 C. x < 3 D. x > 3 E. x < -3

Based on our step-by-step calculation and verification, the correct answer is D. x > 3. This represents all real numbers that are strictly greater than 3. Graphically, this would be an open circle at 3 on a number line with the line extending infinitely to the right.

Key Takeaways for Solving Inequalities

Solving inequalities shares many similarities with solving equations, but it's crucial to remember the distinct rules. The primary goal remains isolating the variable. You can add or subtract any number from both sides, and multiply or divide by any positive number without changing the inequality's direction. The critical difference arises when you multiply or divide by a negative number – in such cases, the inequality sign must be flipped. This fundamental rule ensures that the relationship between the two expressions remains valid.

For instance, if we had an inequality like -2x > 6, dividing by -2 would require us to reverse the sign: x < -3. Failing to do so would lead to an incorrect solution. Always double-check your operations, especially when negatives are involved. Practicing various inequality problems will help solidify your understanding of these rules and build confidence in your ability to solve them accurately. Websites like [Khan Academy](https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86: inequalities) offer excellent resources and practice exercises for mastering this topic.

Conclusion

In summary, we successfully solved the inequality 5x - 2 > x + 10 by applying algebraic principles. By isolating the variable 'x' through a series of valid operations, we determined that the solution set includes all numbers strictly greater than 3. This result was confirmed through verification with test values. The correct answer among the given options is x > 3.

For further practice and understanding of linear inequalities, exploring resources like Math is Fun can be very beneficial.