Inequalities are a part of many math problems. Inequality solving is the process of finding the solution set, or range, of numbers that solve a given inequality.
An inequality is a statement containing an operator that does not represent equality. For example, means greater than, and = means equal to.
When solving inequalities, it is important to recognize what part of the statement represents the operator and what parts represent the values that make up the solution set.
When figuring out the solution set for an inequality, you can use the number line to your advantage. This article will discuss how to use the number line to solve inequalities and when it is appropriate to do so.
Determine what each number represents on the number line
The first number on the number line represents the solution of the inequality when x=0. In this case, the solution set is {0}, as 0 is an element of the solution when x=0.
The last number on the number line represents infinity, which means there is no value of x that makes the inequality true. In this case, there is no real number that makes the inequality true- it is just a way to represent that fact.
The middle numbers on the number line represent values of x that make the inequality true. In this case, all numbers greater than or equal to 4 make the inequality true. There are no negative numbers that satisfy this inequality.
–X ≥ 4 translates to X ≤ -4
In this inequality, the variable X represents the number of minutes a person exercises per day. The letter “X” stands for an unknown quantity, which is why this equation is called an inequality.
The symbol ≥ means greater than or equal to, so this part of the equation says that X must be greater than -4 minutes per day.
The symbol −4 means -4 minutes, so this part of the equation says that X must be less than -4 minutes per day.
This graph represents the number line that includes the solution set for this inequality. The light blue area on the graph shows all of the possible values for X that satisfy the inequality -X ≥ 4.Only numbers greater than or equal to -4 are included in this area.
The solution set is all real numbers less than or equal to -4
Inequalities are a part of every day life. Whether it is talking about a fair wage or how much money you have in the bank, everything can be discussed in terms of inequalities.
Inequalities are situations where one variable is greater than or less than another variable. These variables can be numbers, measures, or even states of being. For example, the inequality “taller than” means that the person or thing is taller than something else.
The number line that represents the solution set for the inequality -X ≥ 4 is the left side of the number line. Only numbers less than -4 are solutions to the inequality -X ≥ 4. Any number that is greater than -4 will not solve the inequality.
Watch out for false solutions! Even though 5 is not negative, it is not a solution because it is not less than -4.
The solution set does not include -4 or 6
In this problem, the solution set does not include the numbers -4 or 6. This is due to the fact that when x is greater than or equal to -4 or 6, then the inequality is not true.
For example, if x was -4, then 4 would be greater than x, so x would not be included in the solution set. The same goes for 6, if x was 6 then x would be less than 8, so again it would not be included in the solution set.
The reason why these numbers are not in the solution set is because they break down the inequality. In other words, they prove that the inequality is not true for at least one value of x.
There are many more ways to prove an inequality is true or false, and this problem specifically uses counting proofs.
Check all points on the line to make sure they work
Now let’s look at the line that goes through the points (–4,4), (0,4), (4,0), and (5,–1). All of these points are on the line 4 ≥ –X, so they are valid solutions to this inequality.
Now let’s look at the line that goes through the points (−2,-1), (0,-1), (-1,0) and (-2,3). All of these points are on the line –X ≥ 4, so they are valid solutions to this inequality.
Notice that both lines pass through the point (–2,-1). You could test all of the points on either line to make sure they were valid solutions. Both lines have the same number of solutions, though- four.
There are more than one solution for each inequality depending on which side of the line(s) you find them.
The line represents all of the points in the solution set with perfect precision
In this case, the solution set is all numbers greater than or equal to 4. So, in other words, the solution set is all numbers 4 or less.
The number line works because whenever you drew a line parallel to the bottom number line and touched the top number line, it represented a true statement: All numbers less than or equal to 4 are less than or equal to 5.
Any time you drew a line parallel to the bottom number line and did not touch the top number line, it did not represent a true statement. In fact, it represented a false statement: It said that all of those numbers were greater than 5, but that is not true! Only numbers greater than 4 are greater than 5.
Convert the inequality into an equation if possible
If the inequality cannot be solved as is, you can try to solve it as an equation. To do this, convert the inequality into an equation using the equals sign.
For example, if the inequality is X ≥ 4, then X equals 4 is the solution. You can test this solution by checking whether X is greater than or equal to 4.
If X is less than 4, then the solution is not a valid solution and must be re-evaluated. If X is equal to 4, then the solution is valid and the problem is solved!
This process can be difficult so it is best to know how to do it before encountering a problem like this one. Knowing how to convert an inequality into an equation will help with this task.
