We have learned that equations are statemtents that tell us that two expressions are equal. An inequality, on the other hand, relates two expressions that are not equal. We use symbols like , ≥, ≤, and ≠ signs to establish a relation between two expressions. . Read More Read Less
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The term inequality refers to a mathematical expression whose values to the left and the right side are not equal. In a nutshell, an inequality compares any two values and shows that one is less than, greater than or not less than and not greater than the other quantity. Inequalities are used to compare numbers and find the range of values that satisfy a variable’s conditions.
Fig: shows Economic inequality
In general, five inequality symbols are used to represent inequality equations. Less than (<), greater than (>), less than or equal \((\leq )\) and greater than or equal \((\geq )\) are some of the symbols used to represent inequalities.
For example, writing sentences as inequalities.
A number that, when substituted for x, produces a true statement in an inequality is a solution for the inequality in x. The set of all solutions for an inequality is known as the solution set. In most cases, an inequality has an infinite number of solutions, which can be easily described using an interval notation.
For example, We have to tell whether the given value of x is the solution of the inequality given. \(x+4>10 ;~x=5 \)
\(x+4>10 \) (Writing the inequality)
\(5+4>10 \) (Substituting 5 in the place of x)
Hence, 5 is not a solution to the inequality.
An open circle is used when that number is not a solution of the inequality and a closed circle is used when the number is a solution of the inequality. In other words, use an open circle for ‘less than’ or ‘greater than’, and a closed circle for ‘less than or equal to’ or ‘greater than or equal to’ when graphing a linear inequality on a number line. A left arrow represents that the solutions extend to the left, and a right arrow represents that the solutions extend to the right. The arrows are shaded and every number on the shaded region is a solution to the inequality.
All values of \(x \) that are only less than 4 will be the solution to this problem (not equal to 4). All values that are graphed to the left of 4 but not four will be the solution. An open circle for 4 and an arrow extending to the left will be used in the graph to indicate that 4 is not included in the solution.
In the shaded region, a number chosen makes the inequality true. In the non-shaded area, a number chosen makes the inequality false.
The shaded arrow pointing left denotes that the shaded arrow will continue to the left indefinitely.
Also, graphing the solution set of \(x\geq 3 \) .
All values of \(x \) greater than or equal to 3 will be the solution to this problem. All values graphed to the right of 3 will be the solution, including 3. Because \(x~=~3 \) is one of the solutions, the graph will use a closed circle with an arrow extending to the right.
In the shaded region, a number chosen equals true. In the non-shaded area, a number chosen equals false.
Picking 0: 0 > 3 FALSE
Picking 1: 1 > 3 FALSE
Picking 2: 2 > 3 FALSE
Picking 4: 4 > 3 TRUE
The shaded arrow pointing right denotes that the shaded arrow will continue to the right indefinitely.
