To solve linear inequalities in one step, isolate the variable by performing the inverse operation on both sides of the inequality. Remember that if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
Here's a breakdown with examples:
1. Understand the Goal:
The goal is to get the variable (e.g., x, y, z) by itself on one side of the inequality. This is called isolating the variable.
2. Identify the Operation:
Determine what operation is being performed on the variable. Common operations include addition, subtraction, multiplication, and division.
3. Perform the Inverse Operation:
Do the opposite of the operation being performed.
- If adding: Subtract from both sides.
- If subtracting: Add to both sides.
- If multiplying: Divide both sides.
- If dividing: Multiply both sides.
4. The Negative Number Rule:
- Crucially: If you multiply or divide both sides of the inequality by a negative number, you must reverse the direction of the inequality sign.
5. Simplify:
Simplify both sides of the inequality to obtain the solution.
Examples:
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Example 1: Addition
- Inequality:
y + 7 < 8 - Operation: Adding 7 to y
- Inverse Operation: Subtract 7 from both sides.
y + 7 - 7 < 8 - 7y < 1- Solution: y is less than 1
- Inequality:
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Example 2: Subtraction
- Inequality:
x - 3 ≥ 5 - Operation: Subtracting 3 from x
- Inverse Operation: Add 3 to both sides.
x - 3 + 3 ≥ 5 + 3x ≥ 8- Solution: x is greater than or equal to 8.
- Inequality:
-
Example 3: Multiplication
- Inequality:
2z > 10 - Operation: Multiplying z by 2
- Inverse Operation: Divide both sides by 2.
2z / 2 > 10 / 2z > 5- Solution: z is greater than 5.
- Inequality:
-
Example 4: Division
- Inequality:
x / 4 ≤ 3 - Operation: Dividing x by 4
- Inverse Operation: Multiply both sides by 4.
(x / 4) * 4 ≤ 3 * 4x ≤ 12- Solution: x is less than or equal to 12.
- Inequality:
-
Example 5: Multiplication/Division by a Negative Number
- Inequality:
-3y ≤ 12 - Operation: Multiplying y by -3
- Inverse Operation: Divide both sides by -3.
-3y / -3 ≥ 12 / -3(Notice the inequality sign flipped!)y ≥ -4- Solution: y is greater than or equal to -4
- Inequality:
In summary, solving one-step linear inequalities involves isolating the variable by using inverse operations, paying close attention to reversing the inequality sign when multiplying or dividing by a negative number.
