What makes an inequality all real numbers

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 Inequalities with No Solution or Infinite Solutions. Go to Topic. Explanations 3. Hannah Bonville. Here are a couple graphics to get you thinking before looking at some more complex examples. Image source: By Caroline Kulczycky. Inequalities with no solution What you're looking for in this case is nonsense. Let's look at a few examples.

Show Solution Check. Related Lessons. Whether the input was positive or negative or zero , the output is always positive or zero. Step 3: Solve for the unknown in both equations. Step 4: Check your answer analytically or graphically. Asked by: Daren Turanov asked in category: General Last Updated: 4th June, How do you know if an absolute value inequality is all real numbers?

The absolute value of any number is either zero 0 or positive. It makes sense that it must always be greater than any negative number. The answer to this case is always all real numbers. What is a no solution inequality? If the inequality states something untrue there is no solution. Register Login Username. What is the difference between an 'All real numbers' absolute value inequality and a 'No solution' inequality?

Nirvana Aug 30, CPhill Aug 30, Ahh that makes more sense!!