Solving Inequalities: 2x + 9 ≤ 3 - x ≤ 6x + 17
Inequalities are an essential concept in algebra, and they can be quite challenging to solve. In this article, we will explore how to solve the inequality 2x + 9 ≤ 3 - x ≤ 6x + 17.
Step 1: Simplify the Inequality
The first step in solving this inequality is to simplify it by combining the two inequalities into a single one. We can do this by using the following rule:
If a ≤ b and b ≤ c, then a ≤ c
Using this rule, we can rewrite the inequality as:
2x + 9 ≤ 6x + 17
Step 2: Isolate the Variable x
Now, our goal is to isolate the variable x. We can do this by subtracting 2x from both sides of the inequality:
9 ≤ 4x + 17
Next, we can subtract 17 from both sides:
-8 ≤ 4x
Finally, we can divide both sides by 4 to solve for x:
-2 ≤ x
Step 3: Write the Solution in Interval Notation
The solution to the inequality is x ≥ -2. We can write this in interval notation as:
[-2, ∞)
This means that all values of x greater than or equal to -2 are solutions to the inequality.
Conclusion
In this article, we have solved the inequality 2x + 9 ≤ 3 - x ≤ 6x + 17. We simplified the inequality, isolated the variable x, and wrote the solution in interval notation. By following these steps, we can solve similar inequalities and better understand the concepts of algebra.
