- 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)

- We write inequalities when we don't have any exact solution of the given problems.
- Just like equations, we can write inequalities on the basis of the given statements.

**For example:**

The sum of \(x\) and \(10\) is less than \(20\).

The statement says, sum of \(x\) and \(10\).

\(\Rightarrow\;x+10\)

\(x+10\) is less than \(20\).

\(\Rightarrow\;x+10<20\)

A \(\left(\dfrac{1}{3}\right)n>3+15n\)

B \(3n-3>15\)

C \(\left(\dfrac{1}{3}\right)n+3<15\)

D \(3n+3<15\)

- Solving an inequality means to find all the possible values that represent the solution of the given problem.
- To solve an inequality, do the same thing on both sides of the inequality.
- Use inverse operations to solve an inequality.

**For example:**

\(x+5\leq13\)

- To solve this inequality, subtract \(5\) from both the sides.

\(\Rightarrow\;x+5-5\leq13-5\)

\(\Rightarrow\;x\leq8\)

This is the required solution.

A \(x>2\)

B \(x<1\)

C \(x>1\)

D \(x\leq2\)

- 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.

- We write inequalities when we don't have any exact solution of the given problems.
- Just like equations, we can write inequalities according to the statements.
- To write inequalities for word problems, first identify the keywords:

(i) At least - means greater than or equal to \((\geq)\)

(ii) Maximum or not more than - means less than or equal to \((\le)\)

(iii) More than - means greater than \((>)\)

(iv) Less than - means less than \((<)\)

- To write an inequality for word problem, follow the given steps:

- Try to understand what the situation says.
- Find the operations involved.
- Identify the keyword used in the problem to determine the inequality symbol.

A \(x>0\)

B \(x<0\)

C \(x>5\)

D \(x<2\)

- 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 \(h\leq 2\)

A \(m<300\)

B \(m>300\)

C \(m\ge 300\)

D \(m\le300\)

- We write inequalities when we don't have any exact solution of the given problem.
- To write inequalities for word problems, identify the keywords:

(i) at least - means greater than or equal to \((\geq)\)

(ii) maximum or not more than - means less than or equal to \((\leq)\)

(iii) more than - means greater than \((>)\)

(iv) less than - means less than \((<)\)

- To write an inequality for word problems, follow the given steps:

- Try to understand what the situation says.
- Find the operations involved.
- Identify the keyword used in the given problem to determine the inequality symbol.

**For example:**

A cab charges \($2\) flat rate in addition to \($0.50\) per mile. Tina has no more than \($10\) to spend on a ride.

- To represent Tina's situation, go step by step according to the statement.
- Let the number of miles she can travel \(=x\)

\(\therefore\) Ride charges \(=0.50x+2\)

- She has no more than \($10\) to spend.

Here, the keyword is "no more than" i.e. \((\leq)\)

\(\Rightarrow\;0.50x+2\leq10\)

A \(x>4\)

B \(x<4\)

C \(x\geq4\)

D \(x\leq4\)