What is an inequality when is it used

Whether a sign is greater than or less than depends on the direction of the inequality sign. This means that the two sides of an inequality expression could potentially be equal. However, not enough is known to prove this. As you can see in the mathematical expression above, x , is greater than or equal to 7. That's why we must use. Whenever a linear inequality has a variable and a real number, you can express it on a number line.

Here's how to use number lines to show x is greater than positive number 3 and less than or equal to negative number When we know an inequality is between two numbers, you can write it in interval notation.

Here's how you would show that y is less than or equal to -4 and 2 is greater than y :. As you can see, we use an open circle to show that y is less than 2 and a closed circle to show that y is equal to or greater than 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. To see how the rules of addition and subtraction apply to solving inequalities, consider the following:. This means that we must also change the direction of the symbol:.

Note that it would become problematic if we tried to multiply or divide both sides of an inequality by an unknown variable. Because the rules for multiplying or dividing positive and negative numbers differ, we cannot follow this same rule when multiplying or dividing inequalities by variables.

Variables can, however, be added or subtracted from both sides of an inequality. A compound inequality involves three expressions, not two, but can also be solved to find the possible values for a variable. There are actually two statements here. For a visualization of this inequality, refer to the number line below.

The numbers 4 and 9 are not included, so we place open circles on these points. Again, because the numbers -2 and 0 are not included, we place open circles on those points.

However, the meaning of this is difficult to visualize—what does it mean to say that an expression , rather than a number, lies between two points? Finally, it is customary though not necessary to write the inequality so that the inequality arrows point to the left i.

However, this is wrong. What numbers work? How about ? By playing with numbers in this way, you should be able to convince yourself that the numbers that work must be somewhere between and This is one way to approach finding the answer. The other way is to think of absolute value as representing distance from 0. Once again, we conclude that the answer must be between and This answer can be visualized on the number line as shown below, in which all numbers whose absolute value is less than 10 are highlighted.

It is not necessary to use both of these methods; use whichever method is easier for you to understand.