Steps to Solve Double Inequalities

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

1. Understand the Inequality
   First, understand the given double inequality. For example, consider: $-3 < 2x + 1 leq 7$

2. 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: $-3 < 2x + 1 leq 7$

1. Subtract 1 from all parts:
   $-3 – 1 < 2x + 1 – 1 leq 7 – 1$

Simplifies to:
$-4 < 2x leq 6$

2. Divide all parts by 2:
   $frac{-4}{2} < frac{2x}{2} leq frac{6}{2}$

Simplifies to:
$-2 < x leq 3$

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.

4. 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)

5. 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

1. Forgetting to Reverse Inequality Signs: When multiplying or dividing by a negative number, always reverse the inequality signs.
2. Incorrect Operations: Ensure you perform the same operation on all three parts of the inequality.
3. 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.

Citations

1. 1. Khan Academy – Solving Inequalities
2. 2. Math is Fun – Solving Inequalities
3. 3. Purplemath – Solving Inequalities