Inequalities are a fundamental concept in algebra and mathematics, used to compare values that are not necessarily equal. Understanding how to solve them is crucial for various mathematical and real-world applications. This guide will walk you through the process of solving inequalities, ensuring you grasp the key principles and techniques involved.
| Symbol | Words | Example |
|---|---|---|
| > | greater than | x + 3 > 2 |
| < | less than | 7x < 28 |
| ≥ | greater than or equal to | 5 ≥ x − 1 |
| ≤ | less than or equal to | 2y + 1 ≤ 7 |
Understanding the Goal: Isolating the Variable
Just like solving equations, the primary goal when solving inequalities is to isolate the variable (usually ‘x’ or ‘y’, but it can be any letter) on one side of the inequality sign. This means we want to manipulate the inequality until it looks something like:
- x < 5
- y ≥ 11
- z > -2
When the inequality is in this form, we say it is “solved,” because we know the range of values that the variable can take to satisfy the inequality.
Example: Solving a Simple Inequality
Let’s start with a straightforward example:
x + 2 > 12
To solve for ‘x’, we need to get ‘x’ by itself on the left side. We can do this by subtracting 2 from both sides of the inequality:
x + 2 – 2 > 12 – 2
Simplifying both sides, we get:
x > 10
This is the solution! It tells us that any value of ‘x’ greater than 10 will make the original inequality true.
The Golden Rules of Solving Inequalities
Solving inequalities is very similar to solving equations, and many of the same operations apply. However, there’s a crucial aspect to remember: the direction of the inequality.
The “direction” refers to which way the inequality symbol “points” (e.g., <, >, ≤, ≥). Certain operations can change this direction, and it’s vital to know when and how.
Operations That Do Not Change the Inequality Direction (Safe Operations):
These operations are safe to use and will not require you to flip the inequality sign:
- Adding or Subtracting a Number: You can add or subtract the same number from both sides of the inequality without changing its direction.
- Multiplying or Dividing by a Positive Number: Multiplying or dividing both sides by the same positive number also keeps the inequality direction unchanged.
- Simplifying a Side: Simplifying one side of the inequality (e.g., combining like terms) does not affect the inequality direction.
Example of Safe Operations:
3x < 7 + 3
First, simplify the right side:
3x < 10
Now, divide both sides by 3 (a positive number):
3x / 3 < 10 / 3
x < 10/3
The inequality direction remained the same throughout the process.
Operations That Do Change the Inequality Direction (Direction-Changing Operations):
These operations require you to reverse the inequality sign:
- Multiplying or Dividing by a Negative Number: If you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality sign. (
<becomes>,>becomes<,≤becomes≥,≥becomes≤). - Swapping Left and Right Hand Sides: If you decide to swap the entire left-hand side with the entire right-hand side of the inequality, you must also reverse the inequality sign.
Example of Direction-Changing Operations:
2y + 7 < 12
Let’s swap the left and right sides to have the variable term on the right:
12 > 2y + 7
Notice how the < sign changed to > when we swapped sides.
Now, let’s continue solving by subtracting 7 from both sides (safe operation):
12 – 7 > 2y + 7 – 7
5 > 2y
Finally, divide both sides by 2 (a positive number – safe operation):
5 / 2 > 2y / 2
5/2 > y
Or, rewriting with ‘y’ on the left (and reversing the sign again to maintain correctness):
y < 5/2
Step-by-Step Techniques for Solving Inequalities
Let’s delve into more specific techniques with examples.
Adding or Subtracting Values in Inequalities
As we saw earlier, adding or subtracting the same value from both sides is a fundamental and “safe” operation.
Example: x + 3 < 7
To isolate ‘x’, subtract 3 from both sides:
x + 3 – 3 < 7 – 3
x < 4
This solution indicates that any value of ‘x’ less than 4 will satisfy the original inequality.
Visual representation of x + 3 < 7 being solved and resulting in x < 4 on a number line.
This works because if one value is less than another, and you add or subtract the same amount from both, the relationship of “less than” remains the same.
Dealing with ‘x’ on the Right Side
Sometimes, after performing operations, your variable might end up on the right side of the inequality. This is perfectly valid, but it’s often conventional to rewrite the solution with ‘x’ on the left. To do this, simply swap sides and remember to reverse the inequality sign.
Example: 12 < x + 5
Subtract 5 from both sides:
12 – 5 < x + 5 – 5
7 < x
This is a correct solution. However, to put ‘x’ on the left, swap sides and reverse the inequality:
x > 7
Both 7 < x and x > 7 represent the same solution set – all numbers greater than 7.
Multiplying or Dividing by Values in Inequalities
Remember the critical rule: multiplying or dividing by a negative number reverses the inequality.
Example: 3y < 15 (Positive Value)
Divide both sides by 3 (positive):
3y / 3 < 15 / 3
y < 5
Example: -2y < -8 (Negative Value)
Divide both sides by -2 (negative) and reverse the inequality:
-2y / -2 > -8 / -2
y > 4
It’s crucial to reverse the inequality sign in the step where you divide by a negative number to get the correct solution.
Caution: When multiplying or dividing inequalities, always pay close attention to the sign of the number you are multiplying or dividing by. Negative numbers require reversing the inequality.
The Danger of Multiplying or Dividing by Variables
Consider this example: bx < 3b
You might be tempted to divide both sides by ‘b’ to get x < 3. However, this is only correct if ‘b’ is a positive number.
- If ‘b’ is positive: Dividing by ‘b’ gives x < 3 (correct).
- If ‘b’ is negative: Dividing by ‘b’ requires reversing the inequality, giving x > 3 (correct in this case).
- If ‘b’ is zero: Division by zero is undefined, and the original inequality becomes 0 < 0, which is false.
Since we don’t know if ‘b’ is positive, negative, or zero, we cannot simply divide by ‘b’ to solve for ‘x’ in this case without further information about ‘b’.
Rule: Avoid dividing or multiplying inequalities by a variable unless you are certain about the variable’s sign (always positive or always negative).
Solving Compound Inequalities (Two Inequalities at Once)
Sometimes, you might encounter inequalities that combine two inequality statements into one, like this:
-2 < (6 – 2x) / 3 < 4
To solve these, you need to apply operations to all three parts of the inequality simultaneously.
Example: -2 < (6 – 2x) / 3 < 4
-
Clear the denominator: Multiply all three parts by 3 (positive, so no sign reversal):
-2 3 < (6 – 2x) / 3 3 < 4 * 3
-6 < 6 – 2x < 12
-
Isolate the term with ‘x’: Subtract 6 from all three parts:
-6 – 6 < 6 – 2x – 6 < 12 – 6
-12 < -2x < 6
-
Isolate ‘x’: Divide all three parts by -2 (negative, so reverse both inequality signs):
-12 / -2 > -2x / -2 > 6 / -2
6 > x > -3
-
Rewrite in standard form: It’s conventional to write compound inequalities with the smaller number on the left. So, reorder the inequality and ensure the signs point correctly:
-3 < x < 6
This solution means ‘x’ is greater than -3 and less than 6, or in other words, ‘x’ is between -3 and 6.
Conclusion: Mastering Inequality Solutions
Solving inequalities follows similar principles to solving equations, with the crucial addition of understanding how operations affect the direction of the inequality sign. Remember these key takeaways:
- Isolate the variable: Your goal is always to get the variable by itself on one side.
- Safe operations: Adding/subtracting any number, multiplying/dividing by a positive number, and simplifying sides do not change the inequality direction.
- Direction-changing operations: Multiplying/dividing by a negative number and swapping sides require reversing the inequality sign.
- Be cautious with variables: Avoid multiplying or dividing by variables unless you know their sign.
- Compound inequalities: Apply operations to all parts of the inequality simultaneously.
By mastering these rules and practicing regularly, you’ll become proficient in solving a wide range of inequalities.
Less Than or Greater Than Inequalities Solving Inequality Word Questions Graphing Linear Inequalities Inequality Grapher
