Solving inequalities using multiplication. And division
To solve inequalities using multiplication and division, you need to keep in mind that the direction of the inequality symbol can change depending on whether you are multiplying or dividing by a positive or negative number. Here are the steps to solve inequalities using multiplication and division:
1. Simplify the inequality if necessary. Simplify both sides of the inequality so that you have a single variable on one side and a constant on the other side.
2. If you are multiplying or dividing both sides of the inequality by a positive number, the direction of the inequality remains the same. For example, if you have "x < 5" and you multiply both sides by 2, you get "2x < 10".
3. If you are multiplying or dividing both sides of the inequality by a negative number, the direction of the inequality changes. For example, if you have "x > 5" and you multiply both sides by -2, you need to flip the inequality symbol to get "-2x < -10". In this case, you also need to remember to reverse the order of the inequality, so it becomes "x > -5".
4. After multiplying or dividing, simplify the inequality if necessary.
5. If you are multiplying or dividing both sides of the inequality by a variable, make sure to consider the possible values of the variable. If the variable can be positive or negative, you will need to split the inequality into two separate cases and solve each case individually.
Always remember to check your solution by plugging it back into the original inequality to ensure it is true.
To solve inequalities using multiplication and division, follow these steps:
1. Identify the inequality symbol (>, <, ≥, ≤) and rewrite the inequality if necessary, so the variable is on the left side of the inequality symbol.
2. If the inequality symbol is ">", "<", or "≥", divide or multiply both sides of the inequality by a positive number. Make sure to keep the inequality symbol pointing in the same direction.
For example, if you have the inequality 2x > 6, you can divide both sides by 2 to get x > 3.
3. If the inequality symbol is "<" or "≤", divide or multiply both sides of the inequality by a negative number. In this case, you need to flip the direction of the inequality symbol.
For example, if you have the inequality -3x < 9, you can divide both sides by -3 and reverse the inequality to get x > -3.
4. Simplify the inequality by performing any necessary calculations on both sides.
For example, if you have the inequality 4x - 2 < 10, you can add 2 to both sides to get 4x < 12. Then divide both sides by 4 to get x < 3.
5. Write the final solution in interval notation or using the inequality symbol.
For example, if your solution is x > 3, you can write it in interval notation as (3, ∞).
These steps can be applied to inequalities involving both multiplication and division to solve for the variable.
Solving inequalities using multiplication and division involves applying the same operations to both sides of the inequality sign in order to solve for the variable.
When using multiplication or division to solve inequalities, it is important to keep in mind the following rules:
1. Multiplying or dividing both sides of an inequality by a positive number does not change the direction of the inequality. For example, if you have the inequality 2x > 4, you can divide both sides by 2 to find x > 2.
2. Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality. For example, if you have the inequality -3x > 9, you can divide both sides by -3, but remember to reverse the inequality sign, giving you x < -3.
Now, let's go through an example to illustrate how to solve an inequality using multiplication and division:
Example:
Solve the inequality 5x < 15.
1. Start by dividing both sides of the inequality by 5 since we want to isolate the variable x.
(5x)/5 < 15/5
x < 3
Therefore, the solution to the inequality 5x < 15 is x < 3.
To check if your solution is correct, you can substitute a value for x, such as 2, into the original inequality and see if it holds true: 5(2) < 15? (10 < 15). Since 10 is indeed less than 15, the solution x < 3 is valid.
Keep in mind that when multiplying or dividing by a negative number, you need to reverse the direction of the inequality. For example, if you have the inequality -2x > 8, you would divide both sides by -2 and reverse the inequality sign to find x < -4 as the solution.
Remember to follow these rules when solving inequalities using multiplication and division, and always double-check your solutions by substituting values back into the original inequality.