There is one very important exception to the rule that multiplying or dividing an inequality is the same as multiplying or dividing an equation. Whenever you multiply or divide an inequality by a negative number, you must flip the inequality sign. One very important implication of this rule is: You cannot divide by an unknown i. In contrast to strict inequalities, there are two types of inequality relations that are not strict:.
In addition to showing relationships between integers, inequalities can be used to show relationships between variables and integers.
For a visualization of this, see the number line below:. Note that an open circle is used if the inequality is strict i. Likewise, inequalities can be used to demonstrate relationships between different expressions.
One useful application of inequalities such as these is in problems that involve maximum or minimum values. Jared has a boat with a maximum weight limit of 2, pounds. He wants to take as many of his friends as possible onto the boat, and he guesses that he and his friends weigh an average of pounds. How many people can ride his boat at once? To see why this is so, consider the left side of the inequality. There are steps that can be followed to solve an inequality such as this one.
For now, it is important simply to understand the meaning of such statements and cases in which they might be applicable. As long as the same value is added or subtracted from both sides, the resulting inequality remains true. Take note that multiplying or dividing an inequality by a negative number changes the direction of the inequality.
In other words, a greater-than symbol becomes a less-than symbol, and vice versa. This statement also holds true. This demonstrates how crucial it is to change the direction of the greater-than or less-than symbol when multiplying or dividing by a negative number.
Solving an inequality that includes a variable gives all of the possible values that the variable can take that make the inequality true. To solve an inequality means to transform it such that a variable is on one side of the symbol and a number or expression on the other side.
Often, multiple operations are often required to transform an inequality in this way. Subtract 6 from both sides to get:. Now, you need to rewrite this expression as a compound inequality. The output of an absolute value expression is always positive, but the " x " inside the absolute value signs might be negative, so we need to consider the case when x is negative. That gives us our two inequalities or our "compound inequality". We can easily solve both of them.
These kinds of problems take some practice, so don't worry if you aren't getting it at first! Keep at it and it will eventually become second nature. You also often need to flip the inequality sign when solving inequalities with absolute values. How to Divide Negative Fractions. View All Related Lessons. Caroline K. Multiplying and Dividing by a Negative to Solve an Inequality When we multiply or divide by a negative to solve an inequality, it's the same as multiplying or dividing by a negative to solve a regular equation!
Multiplying by a Negative. Show Solution Check. Ben Ferrara. Summary We can multiply and divide to solve inequalities just like we do with equations. Whenever we multiply or divide by a negative number, we have to flip the inequality sign. You'll need to keep track of whether you're dividing by a positive or negative number. You've reached the end. How can we improve?
0コメント