# Solving linear equations inequalities algebra

To solving linear equations inequalities algebra means to find the value or values of the variable used in it.
Thus;
(i) to solve the inequality 2x + 4 > 6 means to find the variable x.
(ii) to solve the inequality 5 - 6y $\leq$ 3 means to find the value of y and so on.
The following are the rules must be followed for solving a given linear inequality.
Rule 1 : On transferring the positive term (in addition) from one side of an inequality to its other side, the sign of that transferred number changes to negative.
Examples :
4x + 6 > 7  ⇒  4x > 7 - 6,      21 $\geq$ 2x + 8   ⇒  21 - 8 $\geq$ 2x
7x + 4 $\leq$ 18  ⇒  7x $\leq$ 18 - 4 and so on.

Rule 2: On On transferring the negative term (in subtraction) from one side of an inequality to its other side, the sign of that transferred number changes to positive .
Examples :
3x - 6 > 5  ⇒  3x > 5 + 6,      42 $\geq$ 2x - 9   ⇒  42 + 9 $\geq$ 2x
9x - 1 $\leq$ 20  ⇒  9x $\leq$ 20 + 1 and so on.

Rule 3 : If each term of an equation be multiplied or divided by the same positive number, the sign of inequality remains the same.
Case I : If p is positive and x < y
x < y ⇒ px < py      and $\left ( \frac{x}{p} \right )$ < $\left ( \frac{x}{p} \right )$

x > y ⇒ px > py      and $\left ( \frac{x}{p} \right )$ > $\left ( \frac{x}{p} \right )$

x $\leq$ y ⇒ px $\leq$ py      and $\left ( \frac{x}{p} \right )$ $\leq$ $\left ( \frac{x}{p} \right )$

x $\geq$ y ⇒ px $\geq$ py      and $\left ( \frac{x}{p} \right )$ $\geq$ $\left ( \frac{x}{p} \right )$

For example x $\leq$ 5 ⇒ 3x $\leq$ 3 x 5 and      x $\geq$ 5 ⇒ 3x $\geq$ 3 x 5
Rule 4:If each term of an inequality be multiplied or divided by the same negative number, the sign of inequality will flip (reverse).
Case II : If p is negative,
x < y ⇒ -px > -py      and $\left ( \frac{-x}{p} \right )$ > $\left ( \frac{-x}{p} \right )$

x > y ⇒ -px < -py      and $\left ( \frac{-x}{p} \right )$ > $\left ( \frac{-x}{p} \right )$

x $\leq$ y ⇒ -px $\geq$ -py      and $\left ( \frac{-x}{p} \right )$ $\geq$ $\left ( \frac{-x}{p} \right )$

x $\geq$ y ⇒ -px $\leq$ -py      and $\left ( \frac{-x}{p} \right )$ $\leq$ $\left ( \frac{-x}{p} \right )$

Rule 5 : If the sign of the sides of an inequality are positive or negative, then while doing their reciprocals flip(reverse) the sign of inequality.
(i) x > y $\Leftrightarrow \frac{1}{x}$ < $\frac{1}{y}$     (ii) $x \geq y \Leftrightarrow \frac{1}{x} \leq \frac{1}{y}$
(iii) $x \leq y \Leftrightarrow \frac{1}{x} \geq \frac{1}{y}$ and so on.