Solving inequalities using multiplication and division
To solve inequalities using multiplication and division, follow these steps:
1. If the inequality has a variable on both sides of the inequality sign, move all the terms containing the variable to one side of the inequality by adding or subtracting as needed.
2. Identify whether you need to multiply or divide both sides of the inequality by a positive or negative number.
- If you multiply or divide both sides by a positive number, the inequality sign remains the same.
- If you multiply or divide both sides by a negative number, the inequality sign flips.
3. Perform the multiplication or division operation on both sides of the inequality.
4. Simplify the inequality by combining like terms and obtaining a solution.
5. If necessary, express the solution as an interval or set notation.
Note: Whenever you multiply or divide both sides of an inequality by a negative number, remember to flip the inequality sign.
Example 1: Solve the inequality 3x - 6 < 9.
Step 1: Move all terms involving x to the left side: 3x < 9 + 6.
Simplified: 3x < 15.
Step 2: Since the coefficient of x is positive, we can proceed with multiplication or division without flipping the inequality sign.
Step 3: Divide both sides by 3 to isolate x: x < 5.
Step 4: Simplify the inequality: x < 5.
Step 5: The solution can be written in interval notation as (-∞, 5).
Example 2: Solve the inequality -2(4 - x) ≥ 10.
Step 1: Distribute the negative sign inside the parentheses: -8 + 2x ≥ 10.
Step 2: Since the coefficient of x is positive, we can proceed with multiplication or division without flipping the inequality sign.
Step 3: Add 8 to both sides to isolate 2x: 2x ≥ 18.
Step 4: Divide both sides by 2 to solve for x: x ≥ 9.
Step 5: The solution can be written in interval notation as [9, +∞).
Step 1: Start by identifying the inequality. It could be in the form of "greater than" (>), "less than" (<), "greater than or equal to" (≥), or "less than or equal to" (≤).
Step 2: If the inequality sign has a less than or greater than sign, proceed with steps 3 and 4. If the inequality sign has a less than or equal to or greater than or equal to sign, proceed with steps 5 and 6.
Step 3: If the inequality sign is "<" (less than), divide both sides of the inequality by the same positive number.
Step 4: If the inequality sign is ">" (greater than), divide both sides of the inequality by the same positive number. However, since you are dividing by a positive number, the direction of the inequality sign remains the same.
Step 5: If the inequality sign is "≤" (less than or equal to), divide both sides of the inequality by the same positive number.
Step 6: If the inequality sign is "≥" (greater than or equal to), divide both sides of the inequality by the same positive number. However, since you are dividing by a positive number, the direction of the inequality sign remains the same.
Step 7: Simplify both sides of the inequality if possible.
Step 8: Express the solution as an inequality or an interval, depending on the context.
Note: If you divide or multiply both sides of the inequality by a negative number, you need to flip the inequality sign.
To solve inequalities using multiplication and division, you'll follow similar steps as solving equations. However, there is one important point to consider: if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
Here is a step-by-step guide on solving inequalities using multiplication and division:
1. Start by writing down the inequality. For example, let's say we have the inequality 3x + 5 < 14.
2. Simplify the inequality if possible. In this case, the inequality is already simplified.
3. Identify if you need to multiply or divide to isolate the variable. In this example, we need to get rid of the 5 on the left side of the inequality, so we'll subtract 5 from both sides to isolate the variable.
3x + 5 - 5 < 14 - 5
3x < 9
4. To continue isolating the variable, divide both sides of the inequality by the coefficient of the variable. In this case, divide both sides by 3:
(3x) / 3 < 9 / 3
x < 3
5. Finally, write down the solution to the inequality. The solution is x < 3, meaning any value of x that is less than 3 will satisfy the original inequality.
Remember to always check your solution by substituting it back into the original inequality. In this example, if you replace x with any value less than 3, the inequality will hold true.