Solving The Inequality 2 &lt; 4 + 2X &lt; 5X + 5

[Best Answer] Solve the inequality. 2(4 + 2x) ≥ 5x + 5? from brainly.com

Introduction

Inequalities are mathematical expressions that involve greater than, less than, greater than or equal to, or less than or equal to signs. Solving inequalities involves finding the range of values that satisfy the expression. In this article, we will be discussing how to solve the inequality 2 < 4 + 2x < 5x + 5.

Understanding the Problem

The inequality 2 < 4 + 2x < 5x + 5 can be split into two separate inequalities: 2 < 4 + 2x and 4 + 2x < 5x + 5. The solution to the inequality will be the values of x that satisfy both inequalities.

Solving the First Inequality

To solve 2 < 4 + 2x, we need to isolate x on one side of the inequality. We can do this by subtracting 4 from both sides: 2 - 4 < 4 + 2x - 4. This simplifies to -2 < 2x. Dividing both sides by 2 gives us -1 < x.

Solving the Second Inequality

To solve 4 + 2x < 5x + 5, we need to isolate x on one side of the inequality. We can do this by subtracting 2x and 4 from both sides: 4 + 2x - 2x - 4 < 5x + 5 - 2x - 4. This simplifies to 0 < 3x + 1. Subtracting 1 from both sides gives us -1 < 3x. Dividing both sides by 3 gives us -1/3 < x.

Combining the Inequalities

To find the values of x that satisfy both inequalities, we need to find the intersection of the two solution sets. Since -1 < x and -1/3 < x, the solution set is -1 < x < -1/3.

Checking the Solution

To check if our solution is correct, we can substitute a value within the solution set into the original inequality. For example, if we substitute x = -1/2, we get: 2 < 4 + 2(-1/2) < 5(-1/2) + 5. This simplifies to 2 < 3 < 4.5, which is true. Therefore, our solution set is correct.

Graphical Representation

We can also represent the solution set graphically by plotting the two inequalities on a number line and finding the intersection of the shaded regions. The solution set is represented by the shaded region between -1 and -1/3.

Conclusion

In conclusion, solving the inequality 2 < 4 + 2x < 5x + 5 involves splitting the inequality into two separate inequalities, solving each inequality separately, and finding the intersection of the solution sets. The solution set is -1 < x < -1/3. It is important to check our solution by substituting a value within the solution set into the original inequality.