Answer

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**Hint:**To solve such questions first consider both the condition of absolute value. Then add the number $9$ to both the left-hand side and right-hand side of the given equation. Then multiply both LHS and RHS with the number $3$ . Next divide both LHS and RHS with number $2$ . Finally canceling out the common term in numerator and denominator we get the required solution.

**Complete step by step answer:**

Given $\left| {\dfrac{2}{3}x - 9} \right| = 18$

It is asked to find the solution to the given equation.

Write the given equation as shown below, that is,

$\left| {\dfrac{{2x}}{3} - 9} \right| = 18$

First, consider $\dfrac{{2x}}{3} - 9 = 18$

Next add the number $9$ to both the left-hand side and right-hand side of the equation, that is,

$\dfrac{{2x}}{3} - 9 + 9 = 18 + 9$

Simplifying further we get,

$\dfrac{{2x}}{3} = 27$

Multiply both the sides of the above equation with the number $3$, that is

$\dfrac{{2x}}{3} \times 3 = 27 \times 3$

We get

$2x = 81$

Next divide both sides of the equation with the number $2$, that is,

$\dfrac{{2x}}{2} = \dfrac{{81}}{2}$

Canceling the common terms in numerator and denominator we get

$x = \dfrac{{81}}{2}$

Next, consider $\dfrac{{2x}}{3} - 9 = - 18$

Next add the number $9$ to both the left-hand side and right-hand side of the equation, that is,

$\dfrac{{2x}}{3} - 9 + 9 = - 18 + 9$

Simplifying further we get,

$\dfrac{{2x}}{3} = - 9$

Multiply both the sides of the above equation with the number $3$, that is

$\dfrac{{2x}}{3} \times 3 = - 9 \times 3$

We get

$2x = - 27$

Next divide both sides of the equation with the number $2$, that is,

$\dfrac{{2x}}{2} = \dfrac{{ - 27}}{2}$

Canceling the common terms in numerator and denominator we get

$x = \dfrac{{ - 27}}{2}$

**Hence the solution of the given equation are $x = \dfrac{{81}}{2}$ and $x = \dfrac{{ - 27}}{2}$**

**Additional information:**

An equation that can be written in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are real numbers, and $a$ and $b$ are not both zero, is known as a linear equation in two variables $x$ and $y$. Every solution of the equation is a point on the line representing it.

**Note:**One might not forget to consider the modulus, due to which we have to consider two values for the right-hand side. While solving this type of question always start by reducing the left-hand side of the given equation and then solve. To check whether the obtained solution is correct or not, substitute the obtained solution in the given equation.

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