Following four keywords are used while dealing with inequalities:
greater than \((>)\)
less than \((<)\)
greater than or equal to \((\geq)\)
less than or equal to \((\leq)\)
Let's take an example to understand it easily.
Example:
The sum of a number and \(3\) is greater than \(7.\)
\(\to\) Here, 'sum' is indicating the addition operation i.e. \('+'\)
\(\to\) 'greater than' is indicating the inequality which is shown by the symbol, \('>'\)
\(\to\) 'a number' is indicating a variable, let it be \(x.\)
\(\to\) The statement 'The sum of a number and \(3\)', represents an expression, i.e. \(x+3\)
Thus, inequality for the given statement is-
\(x+3>7\)
\(12\) is less than or equal to the difference of a number and \(9.\)
\(\to\) Here, 'less than or equal to' is indicating inequality which is shown by the symbol, \('\leq'\)
\(\to\) 'the difference' is indicating the subtraction operation, i.e. \('-'\)
\(\to\) 'a number' is indicating a variable, let it be \(y.\)
\(\to\) The statement 'the difference of a number and \(9\)' represents an expression, i.e. \(y-9\)
Thus, inequality for the given statement is-
\(12\leq \,y-9\)
A \(x+8\geq15\)
B \(x+8\leq15\)
C \(x+8>15\)
D \(x+8<15\)
For example:
Suppose we have a solution of an inequality as \(x>5\). It means \(x\) can have infinite values that are more than \(5 .\)
For example: Solve the following inequality for \('a'\).
\(3a+5\leq14\)
\(\to\) Here, \(5\) is being added to \(3a\) therefore we will use inverse operation of addition and subtract \(5\) from both the sides of the inequality.
\(\Rightarrow\;3a+5-5\leq14-5\\ \Rightarrow\;3a+0\leq9\\\Rightarrow\;3a\leq9\)
\(\to\) Now, \(3\) is being multiplied by \('a'\) therefore we will use inverse operation of multiplication and divide by \(3\) on both the sides of the inequality.
\(\Rightarrow\;\dfrac{3a}{3}\leq\dfrac{9}{3}\)
\(\Rightarrow\;a\leq3\)
Thus, the solution of the given inequality is:
\(a\leq3\)
A \(y>-2\)
B \(y\geq-2\)
C \(y\leq-2\)
D \(y>2\)
\(\to\) The solution of an inequality does not provide any specific answer. It represents a set of numbers.
\(\to\) To solve an inequality, we use inverse operations.
\(\to\) There is an important rule to solve an inequality.
Reserving the inequality- If we divide or multiply an inequality with any negative integer, the inequality sign changes, because we are switching the signs of the values so we must flip the inequality sign as well.
Alternatively, on a number line, multiplying/dividing by \(-1\) or any negative integer, reflects points through the origin.
For example: \(1<2\;\;\)
\(1 \) is less than \(2\) but if we multiply by \(-1\), then
\((-1)(1)<(2)(-1)\)
\(-1<-2\) which is incorrect
We have to flip the inequality sign also.
\(-1>-2\)
Now, consider an inequality:
\(-x+3<7\)
To find \(x,\) solve the inequality using inverse operations.
\(-x+3-3<7-3\)
\(-x<4\)
\((-1)(-x)>4(-1)\) (Reserving the inequality)
\(x>-4\)
A \(x\leq2\)
B \(x>11\)
C \(x\geq\dfrac{-11}{2}\)
D \(x\leq\dfrac{-11}{2}\)
\(a\neq b\) says that \(a\) is not equal to \(b.\)
For example: \(3\neq5\)
But \(3<5\)
We use the following symbols when a math sentence is not equal:
(i) \(>\, \rightarrow\) greater than
(ii) \(<\, \rightarrow\) less than
(iii) \(\ge\, \rightarrow\) greater than or equal to ("or equal to" part is indicated by the line underneath of the \(>\) symbol)
(iv) \(\leq \, \rightarrow\) less than or equal to ("or equal to" part is indicated by the line underneath of the \(<\) symbol)
\(\to\) For keyword 'at least', inequality symbol \('\geq'\) is used which means 'greater than or equal to'.
\(\to\) For keywords 'maximum' or 'Not more than', inequality symbol \('\leq'\) is used which means 'less than or equal to'.
\(\to\) For keyword 'More than', inequality symbol \('>'\) is used which means 'greater than'.
\(\to\) For keyword 'Less than', inequality symbol \('<'\) is used which means 'less than'.
Example:
A box can contain maximum \(108\) fruits. Jacob wants to store some apples and \(48\) mangoes in that box. Write an inequality for apples and mangoes that Jacob can store.
\(\to\) Let the number of apples \(=A\)
\(\to\) The sum of apples and \(48\) mangoes represents an expression \(=A+48\)
\(\to\) The statement 'box can contain maximum \(108\) fruits' means a maximum number of fruits that can be stored in the box is \(108\) or less than \(108\).
\(\to\) As we know that for the keyword 'maximum', inequality symbol \('\leq'\) is used.
Thus, inequality for this situation is-
\(A+48\leq108\)
A \(x\leq42\)
B \(x\geq42\)
C \(x>42\)
D \(x<42\)
Representations of inequality on the number line:
Case (i) \(x>c\)
\(x>c\) means \(x\) can take all the values greater than \(c\) but not \(c\).
The given number line shows all the possible values which are greater than \(c.\)
Case (ii) \(x<c\)
\(x<c\) means \(x\) can take all the values less than \(c\) but not \(c\).
The number line shows all the possible values which are less than \(c.\)
In cases (i) and (ii), the blank circles on \(c\) represent that \(c\) is not included in the solution.
Case (iii) \(x\geq c\)
\(x\geq c\) means \(x\) can take all the values greater than \(c\). Also, \(c\) is included.
The number line shows all the possible values which are greater than or equal to \(c.\)
Case (iv) \(x\leq c\)
\(x\leq c\) means \(x \) can take all the values less than \(c\). Also, \(c\) is included.
The number line shows all the possible values which are less than or equal to \(c.\)
In cases (iii) and (iv), the shaded circles show that \(c\) is included in the solution.
For example:
Represent \(x\leq7\) on the number line.
\(x\leq7\) means \(x \) can take all the values less than \(7\). Also, \(7\) is included.
The line above the number line shows all the possible values for \(x.\)
The shaded circle over \(7\) shows that \(7\) is also included in the solution.
Here, we will understand them with the help of an example.
In a race, \(10\) kids participated. The length of the race track was about \(2\) miles.
After the completion of the race, it was observed that each kid covered at least \(1.3\) miles.
Here, the number of kids who covered the distance between \(1.3\) miles to \(2\) miles, represents the discrete solution, shown with red points.
Let \(x\) denote the number of kids.
Then \(x>0\) and \(x\leq 10\)
The distance covered by kids represents the continuous solution, as the distance can be in decimals, shown by a red line.
Let \(h\) be the distance covered by kids.
Then, \(h\geq1.3\) and \(h\leq 2\)