**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.(i)Addition ------------------> Subtraction

(ii)Subtraction ---------------> Addition

(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

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