# Solving one-step equations and inequalities

In solving one-step equations and inequalities, the student should know all the rules of inequality. For this you can refer the previous page of ask-math or visit the link given below.
Rules for solving linear equations inequalities

## Steps involved while solving one-step equations and inequalities

1) Isolate the given variable. For this use the opposite operation rule.
(iii)Multiplication ------------> Division
(iv)Division ------------------> Multiplication.
2) If you are dividing or multiply both the sides by negative number then flip the sign.
(i)greater than (>)
becomes less than (<)
(ii)less than (<)
becomes greater than (>)
(iii)greater than and equal to ($\geq$)
becomes less than and equal to ($\leq$)
(iv)less than and equal to ($\leq$)
becomes greater than and equal to ($\geq$)

Examples on solving one-step equations and inequalities

Example 1 : x + 8 > 11
Solution :
x + 8 > 11
Since there is positive 8 so as to isolate x, we will add negative(-8) on both sides
x + 8 - 8 > 11 - 8
∴ x > 3 ( + 8 - 8 = 0)

Example 2 : x - 11 < 5
Solution :
x - 11 < 5
Since there is negative 11 so as to isolate x, we will add positive 11 (+11) on both sides
x - 11 + 11 < 5 + 11
∴ x < 16 ----- ( - 11 + 11 = 0)

Example 3 : x + 72 $\geq$ 65
Solution :
x + 72 $\geq$ 65
Since there is positive 72 so as to isolate x, we will add negative 72 (-72) on both sides
x + 72 - 72 $\geq$ 65 - 72
∴ x $\geq$ -7 ----- ( 72 - 72 = 0)

Example 4 : 5x $\leq$ 65
Solution :
5x $\leq$ 65
Since there is multiplication between 5 and x so to isolate x, we will divide both side by 5
$\frac{5x}{5}$ $\leq$ $\frac{65}{5}$
∴ x $\leq$ 13 ----- ( 5 ÷ 5 = 1)

Example 5 : $\frac{x}{8}$ $\geq$ 6
Solution :
$\frac{x}{8}$ $\geq$ 6
Since there is division between x and 8 so to isolate x, we will multiply both side by 8
$\frac{x}{8}\times 8$ $\geq$ 8 $\times$ 6
∴ x $\geq$ 48 ----- ( 8 ÷ 8 = 1)

Example 6 : -12x $\leq$ 48
Solution :
-12x $\leq$ 48
Since there is multiplication between negative 12 (-12) and x so to isolate x, we will divide both side by (-12)
$\frac{-12x}{-12}$ $\leq$ $\frac{48}{-12}$
According to inequality rule, we have to
flip the inequality sign.
∴ x $\geq$ -4 ----( -12 ÷ -12 =+1 and 48 ÷ (-12)= -4 )

Example 7 : $\frac{x}{-5}$ $\geq$ - 11
Solution :
$\frac{x}{-5}$ $\geq$ - 11
Since there is division between x and (-5) so to isolate x, we will multiply both side by (-5)
$\frac{x}{-5}\times (-5)$ $\geq$ -11 $\times$ (-5)
According to inequality rule, we have to
flip the inequality sign.
∴ x $\leq$ 55 ----- ( (-5) ÷ (-5) = 1 and (-11) X (-5) = +55