Definition of Inequality
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.
- The notation \(x< y \) indicates that x is strictly smaller than y, whereas \(x> y \) indicates that x is strictly larger than y.
- The notation \(x~\geq ~y\) denotes that x is greater than or equal to y, whereas the notion \(x~\leq ~y\) denotes that x is less than or equal to y.
For example, writing sentences as inequalities.
- The value of x is below 6
\(\Rightarrow~x< 6 \)
- The value of M is more than 4
\(\Rightarrow~m > 4 \)
- The value of y+ 4 is greater than or equal to 7
\(\Rightarrow~(y+4)\geq 7 \)
- The value of p+3 is less than or equal to 8
\(\Rightarrow~(p+3)\leq 8 \)
Solution of an Inequality
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)
But \(9\ngtr 10 \)
Hence, 5 is not a solution to the inequality.
Graph of an 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.
For example, graphing the solution set of \(x<4 \).
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.
Picking 2: 2 < 4 TRUE
Picking 1: 1 < 4 TRUE
Picking 3: 3 < 4 TRUE
Picking 5: 5 < 4 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.
Solved Inequality Examples
1. The minimum number of fruits available is 86. Express this statement as an inequality.
Solution: Number of fruits is taken as \(N\) .
The value of \(N\) is greater than or equal to 86
\(\Rightarrow ~N\geq 86\)
2. Write an inequality to represent 4 more than P is at least 8.
Solution: The value of \((p+4)\) is greater than or equal to 8
\(\Rightarrow ~(p+4)\geq 8\)
3. Joey and Rocky are teammates on the same soccer team. Joey scored three goals more than Rocky last Sunday, but their combined score was less than 6 goals. What is the maximum number of goals Joey could score?
Solution: Let’s assume the number of goals Joey scored as J and R for the number of goals scored by Rocky.
We know Joey scored 3 goals more than Rocky, so: \(R+3~=~J\)
We also know that they scored less than 6 goals as a team: \(J+R< 6\)
\(R+(R+3) < 6\) (as we know
\(R+3~=~J\) substituted)
\(2R+3 < 6\) (simplified)
\(2R+3-3 < 6-3\) (Subtracting 3 from both sides)
\(2R < 3\)(simplified)
\(R < 1.5\)
This means that Rocky scored either 0 or 1 goal. Joey scored 3 goals more than Rocky, so he could have scored 3 or 4 goals.
4. I have $80 in my pocket and I want at least $60 to buy a watch. Can I buy the watch? Represent this statement as an inequality expression using a number line.
Solution: Step 1: Draw a number line where each unit is 10$. So, start at the value of $10, $20, $30 and go on up to $100.
Step 2: Place a dot on the boundary number, which is $60. If the statement uses ‘or equal to’ then we have to use a solid dot (closed circle) otherwise use a hollow dot (open circle).
In the statement given ‘at least’ is used. So, plot a solid dot.
Step 3: If the inequality is greater than, shade the number line to the right from $60 to infinity and we have to shade the arrow up to the end (infinite).
Check: As the statement indicates, dollars ≥ $60 from the graph. In the given question the money we have is $80 and this means 80$ ≥ 60$. This means that there is enough money to purchase the watch.
An equation is a statement that ensures that two mathematical expressions have the same value. That is, both sides of the equation are equal to each other.
An inequality, on the other hand, is a statement that uses the symbols > for greater than or < for less than to indicate that one quantity is greater or less than another. So both sides of an inequality are not equal to the other.
Two or more inequalities are joined together with ‘or’ or ‘and’ to form a compound inequality.
For example, \(x> 5\) or \(x< 1\),
\(-1< y< 5\)
In a lot of real-life situations, we have to deal with inequalities.
In fact, we are so used to using inequality applications that we don’t even realize we’re doing algebra. For example,
1. How many liters of gasoline can be put in a car for $50?
2. Is an apartment’s rent reasonable?
3. Is there time before class to go get lunch, eat it and come back?
4. Without going over budget, how much should each family member’s Christmas present cost?
5. It is impossible to test every value in order to check an inequality. So, in each shaded region, check a value to see if it is correct. You can also check to see if a value in each non-shaded region is false.
Check: Using solution sets.
When Rock R = 0, Then Joey J = 3 and R+J=2, and 2<6 is true.
When Rock R = 1, Then Joey J = 3 and R+J=4, and 4<6 is true.
When Rock R = 0, Then Joey J = 4 and R+J=4, and 4<6 is true.
But When Rock R = 2, Then Joey J = 5 and R+J=7, and 7<6 is false.