**12.1 Linear Inequalities**

**12.1.1 Inequalities**

**1.**To write the relationship between two quantities which are not equal, we use the following inequality signs:

> greater than

< less than

≥ greater than or equal to

≤ less than or equal to

**2.**7 > 4 also means 4 < 7. 7 > 4 and 4 < 7 are

**equivalent inequalities**.

**12.1.2**

**Linear Inequalities in One Unknown**

**1.**An inequality in one unknown to the power of 1 is called a

**linear inequality**.

*2*

**Example:***x*+ 5 > 7

**A linear inequality can be represented on a**

2.

2.

**number line**.

**12.1.3**

**Computation on Inequalities**

**1.**When a number is added or subtracted from both sides of an inequality, the condition of the inequality is

**unchanged**.

*Example:*Given 5 > 3

Then, 5 + 2 > 3 + 2 ← (symbol ‘>’ remains)

Hence, 7 > 5

**2.**When both sides of an inequality are multiplied or divided by the same positive number, the condition of the inequality is

**unchanged**.

*Example:* Given 4

*x*≤ 16Then, 4

*x*÷ 4 ≤ 16 ÷ 4 ← (symbol ‘≤’ remains) Hence,

*x*≤ 4**3.**When both sides of an inequality are multiplied or divided by the same negative number, the inequality is

**reversed**.

*Example:*Given –3 > –5

Hence,

**3 < 5**Given –5

*y*> –10Then, –5

*y*÷ 5 > –10 ÷ 5 –

*y*> –2Hence

**,***y*< 2**12.1.4**

**Solve Inequalities in One Variable**

To solve linear inequalities in one variable, use inverse operation to make the variable as the subject of the inequality.

*Example:*Solve the following linear inequalities.

$\begin{array}{l}\text{(a)}3-2x<1\\ \text{(b)}\frac{5-2x}{3}\le 7\end{array}$

*Solution:*(a)

$\begin{array}{l}\text{}3-2x<1\\ 3-2x-3<1-3\\ \text{}-2x<-2\\ \text{}2x>2\\ \text{}\frac{2x}{2}>\frac{2}{2}\\ \text{}x>1\end{array}$

(b)

$\begin{array}{l}\text{}3-2x<1\\ 3-2x-3<1-3\\ \text{}-2x<-2\\ \text{}2x>2\\ \text{}\frac{2x}{2}>\frac{2}{2}\\ \text{}x>1\end{array}$

(b)

**12.1.5**

**Simultaneous Linear Inequalities in One Variable**

**1.**The

**common values**of two simultaneous inequalities are values which satisfy both linear inequalities.

The common values of the simultaneous linear inequalities

*x*≤ 3 and*x*> –1 is –1 <*x*≤ 3.**2.**To solve two simultaneous linear inequalities is to find a

**single equivalent inequality**which satisfies both inequalities.