Solving Inequalities: 6 - x 15 or 2x - 9 ≥ 3
In this article, we will explore how to solve a compound inequality involving two expressions: 6 - x 15 or 2x - 9 ≥ 3. We will break down the steps to simplify and solve the inequality, and provide explanations for each step.
Step 1: Simplify the First Expression
The first expression is 6 - x 15. To simplify this expression, we need to follow the order of operations (PEMDAS):
6 - x 15 = 6 - (x × 15)
= 6 - 15x
Step 2: Simplify the Second Expression
The second expression is 2x - 9 ≥ 3. To simplify this expression, we can add 9 to both sides of the inequality:
2x - 9 + 9 ≥ 3 + 9
2x ≥ 12
Step 3: Divide Both Sides by 2
To get x by itself, we can divide both sides of the inequality by 2:
(2x)/2 ≥ 12/2
x ≥ 6
Solving the Compound Inequality
Now we have two simplified expressions: 6 - 15x and x ≥ 6. To solve the compound inequality, we need to find the values of x that satisfy both expressions.
Method 1: Graphical Method
We can graph both expressions on the number line and find the overlap between the two.
The graph of 6 - 15x is a downward-sloping line, while the graph of x ≥ 6 is a horizontal line.
The overlap between the two graphs is the region where x ≥ 6 and y ≤ 6 - 15x. This region satisfies both expressions.
Method 2: Algebraic Method
We can solve the compound inequality algebraically by combining the two expressions:
6 - 15x ≥ 3 or x ≥ 6
To solve this compound inequality, we can find the values of x that satisfy both inequalities.
Let's solve the first inequality:
6 - 15x ≥ 3
Subtract 6 from both sides:
-15x ≥ -3
Divide both sides by -15:
x ≤ 0.2
Now, let's solve the second inequality:
x ≥ 6
Combining the two solutions, we get:
x ≥ 6 or x ≤ 0.2
Conclusion
In conclusion, we have solved the compound inequality 6 - x 15 or 2x - 9 ≥ 3 using both graphical and algebraic methods. The solution set is x ≥ 6 or x ≤ 0.2. This inequality has two distinct regions that satisfy both expressions.
