# 12.1 Linear Inequalities

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.
Example: 2x + 5 > 7

2.
A linear inequality can be represented on a number line.
Example:

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 4x ≤ 16
Then, 4x ÷ 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 –5y > –10
Then, –5y ÷ 5 > –10 ÷ 5
–y > –2
Hence, 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)
$\begin{array}{l}\text{}\frac{5-2x}{3}\le 7\\ \left(\frac{5-2x}{3}\right)×3\le 7×3\\ \text{}5-2x\le 21\\ \text{}-2x\le 16\\ \text{}2x\ge -16\\ \text{}\frac{2x}{2}\ge -\frac{16}{2}\\ \text{}x\ge -8\end{array}$

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 inequalitywhich satisfies both inequalities.