Informative line

Inequalities

Writing an Inequality from a Statement

  • Writing an inequality from a given statement is almost similar to writing an equation from a statement.
  • The only difference is the use of different keywords.
  • 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-

\(x+3>7\)

  • Consider another statement:

\(12\) 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-

\(12\leq \,y-9\)

Illustration Questions

Choose the correct inequality for the given statement: The sum of a number and \(8\) is greater than or equal to \(15\).

A \(x+8\geq15\)

B \(x+8\leq15\)

C \(x+8>15\)

D \(x+8<15\)

×

For the given statement,

'Sum' is indicating the addition operation, i.e. \(+\)

'A number' is indicating a variable, say \(x.\)

The statement 'the sum of a number and \(8\)' represents an expression, i.e. \(x+8\)

'Greater than or equal to' is indicating the inequality, shown by the symbol, \(\geq\)

Thus, inequality for the given statement is-

\(x+8\geq15\)

Hence, option (A) is correct.

Choose the correct inequality for the given statement: The sum of a number and \(8\) is greater than or equal to \(15\).

A

\(x+8\geq15\)

.

B

\(x+8\leq15\)

C

\(x+8>15\)

D

\(x+8<15\)

Option A is Correct

Solving Inequality Involving Single Variable

  • Inequalities do not have a unique solution.
  • They can have infinite solutions.

For example:

Suppose we have a solution of an inequality as \(x>5\). It means \(x\) can have infinite values that are more than \(5 .\)

  • Solving an inequality involving single variable means, we have to solve for the value of the variable.
  • To find the value of the variable, separate the variable using inverse operations.

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:

\(a\leq3\)

 

Illustration Questions

Solve the following inequality for \(y:\) ​\(5y+2>12\)

A \(y>-2\)

B \(y\geq-2\)

C \(y\leq-2\)

D \(y>2\)

×

Given inequality: \(5y+2>12\)

Here, \(2\) is being added to \(5y\) therefore we will use inverse operation of addition and subtract \(2\) from both the sides of the inequality.

\(\Rightarrow\;5y+2-2>12-2\)

\(\Rightarrow\;5y+0>10\)

\(\Rightarrow\;5y>10\)

Now, \(5\) is being multiplied by \(y\) therefore we will use inverse operation of multiplication and divide by \(5\) on both the sides of the inequality.

\(\Rightarrow\;\dfrac{5y}{5}>\dfrac{10}{5}\)

Thus, the solution of the given inequality is \(y>2\).

Hence, option (D) is correct.

Solve the following inequality for \(y:\) ​\(5y+2>12\)

A

\(y>-2\)

.

B

\(y\geq-2\)

C

\(y\leq-2\)

D

\(y>2\)

Option D is Correct

Reversing the Inequality

\(\to\) 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)

\(x>-4\)

 

 

Illustration Questions

\(-2x-3\leq8\) Solve for \(x.\)

A \(x\leq2\)

B \(x>11\)

C \(x\geq\dfrac{-11}{2}\)

D \(x\leq\dfrac{-11}{2}\)

×

Given:

\(-2x-3\leq8\)

Adding \(3\) to both the sides of the inequality because addition is the inverse operation of subtraction.

\(-2x-3+3\leq8+3\)

\(-2x\leq11\)

Dividing by \(-2\) on both the sides of the inequality because division is the inverse operation of multiplication.

\(\dfrac{-2x}{-2}\geq\dfrac{-11}{2}\)  (Reversing the inequality)

\(x\geq\dfrac{-11}{2}\)

Hence, option (C) is correct.

\(-2x-3\leq8\) Solve for \(x.\)

A

\(x\leq2\)

.

B

\(x>11\)

C

\(x\geq\dfrac{-11}{2}\)

D

\(x\leq\dfrac{-11}{2}\)

Option C is Correct

Inequality

  • An inequality says that the two values are not equal.

\(a\neq b\) says that \(a\) is not equal to \(b.\)

  • Sometimes, we come across such situations where two quantities are not equal \((\neq)\). One may be greater than or less than the other one.

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)

Illustration Questions

Which one of the following shows an inequality?

A \(x+1=2\)

B \(2x+3\)

C \(x+y=4\)

D \(x>7\)

×

An inequality says that the two values are not equal, which means one may be greater than (>) or less than (<) the other one.

Option (A) is  \(x+1=2\) which represents an equation, because it contains an equals (=) sign.

Hence, option (A) is incorrect.

Option (B) is  \(2x+3\) which represents an expression, because it does not contain an equals sign.

Hence, option (B) is incorrect.

Option (C) is \(x+y=4\) which represents an equation, because it contains an equals sign.

Hence, option (C) is incorrect.

Option (D) is \(x>7\) which represents an inequality as it says that \(x\) is greater than \(7.\)

Hence, option (D) is correct.

Which one of the following shows an inequality?

A

\(x+1=2\)

.

B

\(2x+3\)

C

\(x+y=4\)

D

\(x>7\)

Option D is Correct

Writing Inequalities from Real Situation

  • Writing an inequality from a given real situation is almost similar to writing an equation from a real situation.
  • Keywords are the base to write an inequality from a real situation.
  • Different inequality symbols are used corresponding to different keywords as discussed below:

\(\to\) 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'.

  • Following two points to be considered for writing an inequality.
  1. Identify how the given quantities are related to each other.
  2. Use the appropriate inequality symbol.

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-

\(A+48\leq108\)

Illustration Questions

Cody wants to buy \(2\) shirts from a store. The minimum cost of one shirt in the store is \($42\). How much money should Cody have so that he can purchase at least one shirt? Write an inequality.

A \(x\leq42\)

B \(x\geq42\)

C \(x>42\)

D \(x<42\)

×

Given:

Minimum cost of one shirt in the store \(=$42\)

Let the money Cody should have \(=x\)

The minimum cost of one shirt is \($42\) which means that the cost of each shirt is \($42\) or more than \($42\).

As we know that for the keyword 'at least', inequality symbol \('\geq'\) is used.

Thus, inequality for this situation is-

\(x\geq42\)

Hence, option (B) is correct.

Cody wants to buy \(2\) shirts from a store. The minimum cost of one shirt in the store is \($42\). How much money should Cody have so that he can purchase at least one shirt? Write an inequality.

A

\(x\leq42\)

.

B

\(x\geq42\)

C

\(x>42\)

D

\(x<42\)

Option B is Correct

Representation of Inequality through Number Line

  • We can represent inequality on the number line to understand it clearly.
  • First, draw a number line and then place the value from where we want to represent.
  • For greater than \((>)\) and greater than or equal to \((\geq)\) inequalities, go to the right side from the value on the number line.
  • For less than \((<)\) and less than or equal to \((\leq)\) inequalities, go to the left side from the value on the number line.

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.

Illustration Questions

Which option correctly represents the given inequality: \(x\geq5\)?

A

B

C

D

×

Given inequality: \(x\geq5\) is in the form of \(x\geq c\).

\(x\geq5\) means \(x\)  can take all the possible values which are greater than or equal to \(5.\) 

For \(x \) greater than \(5\), we go to the right side from \(5\) on the number line.

As \(5\) is also included in the solution, so we can show it by placing a shaded circle over \(5\).

 

Now, we can represent it through a number line as shown.

image

Hence, option (B) is correct.

Which option correctly represents the given inequality: \(x\geq5\)?

A image
B image
C image
D image

Option B is Correct

Discrete and Continuous Solution of Inequality

  • An inequality can have infinite solutions. 
  • These solutions can be discrete or continuous. 
  • But, what are discrete and continuous solutions?

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 \(h\leq 2\)

Illustration Questions

In a school, the height of students of grade \(1\) is between \(35\) inches to \(45\) inches. Which one of the following options represents the height of students of grade \(1\)?

A

B

C

D

×

The height of students of grade \(1\) is between \(35\) inches to \(45\) inches.

The height of students is represented by red color on the number line shown. The numbers \(35\) and \(45\) are included. It shows continuous solution because height can be anywhere between \(35\) to \(45\) inches and can be in decimals.

Negative numbers are not shown on the number line because height can not be negative.

Hence, option (A) is correct. 

image

Option (B) is incorrect because it does not represent all possible heights.

Option (C) is incorrect because it represents only three heights, \(35\) inches, \(40\) inches and \(45\) inches.

Option (D) is incorrect because it represents only the heights between \(40\) and \(45\) inches.

In a school, the height of students of grade \(1\) is between \(35\) inches to \(45\) inches. Which one of the following options represents the height of students of grade \(1\)?

A image
B image
C image
D image

Option A is Correct

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