Double inequalities are expressions that involve two inequality signs. For example, $a < x < b$. Solving double inequalities involves finding the range of values that satisfy both inequalities simultaneously. Let’s go through the steps to solve them.
Step-by-Step Process
- Understand the Inequality
First, understand the given double inequality. For example, consider: $-3 < 2x + 1 leq 7$
- Isolate the Variable
The goal is to isolate the variable in the middle. To do this, perform the same mathematical operations on all three parts of the inequality.
Example
Given: $-3 < 2x + 1 leq 7$
- Subtract 1 from all parts:
$-3 – 1 < 2x + 1 – 1 leq 7 – 1$
Simplifies to:
$-4 < 2x leq 6$
- Divide all parts by 2:
$frac{-4}{2} < frac{2x}{2} leq frac{6}{2}$
Simplifies to:
$-2 < x leq 3$
Graph the Solution
Graphing the solution on a number line helps visualize the range of values that satisfy the inequality. For $-2 < x leq 3$:- Draw an open circle at -2 (indicating -2 is not included).
- Draw a closed circle at 3 (indicating 3 is included).
- Shade the region between -2 and 3.
- Verify the Solution
Pick a value within the solution range and substitute it back into the original inequality to verify.
Example
For $x = 0$ (which is between -2 and 3):
$-3 < 2(0) + 1 leq 7$
$-3 < 1 leq 7$ (True)
Write the Solution in Interval Notation
Express the solution in interval notation for clarity. For $-2 < x leq 3$:$(-2, 3]$
Common Mistakes to Avoid
- Forgetting to Reverse Inequality Signs: When multiplying or dividing by a negative number, always reverse the inequality signs.
- Incorrect Operations: Ensure you perform the same operation on all three parts of the inequality.
- Misinterpreting Open and Closed Intervals: Use open circles for $<$ or $>$ and closed circles for $leq$ or $geq$
Conclusion
Solving double inequalities involves isolating the variable by performing consistent operations on all parts of the inequality. Visualizing the solution on a number line and verifying your solution helps ensure accuracy. With practice, solving double inequalities becomes straightforward and intuitive.