Question

Solve $3x + 2 < 5\left( {x - 4} \right)$.

Answer 1

Hint:Now this inequality can be solved in the same manner as that we solve a normal algebraic expression. In order to solve it we need to manipulate the given equation in such a way that we should get $x$ by itself. In order to get $x$ by itself we can perform any arithmetic operations on both LHS and RHS equally at the same time.

Complete step by step answer:
Given, $3x + 2 < 5\left( {x - 4} \right)................................\left( i \right)$.
To solve an inequality simply means to find a range of values that an unknown variable x can take and thus satisfy the inequality.Such that we can proceed in the same manner as that of solving normal algebraic expression.Now we have to manipulate the given equation in terms of only $x$, which can be achieved by performing different arithmetic operations on both LHS and RHS equally.First let’s distribute 5 in the RHS using distributive property:
\[\Rightarrow 3x + 2 < 5\left( {x - 4} \right) \\
\Rightarrow 3x + 2 < 5x - 20.........................\left( {ii} \right) \\ \]
In order to isolate the term$x$alone from (ii) we have to add +20 to the LHS and RHS since by adding +20 to both LHS and RHS we can isolate the term $5x$ and proceed accordingly:
\[\Rightarrow 3x + 2 + 20 < 5x - 20 + 20 \\
\Rightarrow 3x + 22 < 5x.................................\left( {iii} \right) \\ \]
On simplifying (iii) we can write:
$\Rightarrow 22 < 5x - 3x \\
\Rightarrow 2x > 22.................................\left( {iv} \right) \\ $
Now in order to isolate the term $x$ alone we have to divide the LHS and RHS by the number 2 in the equation (iv). Such that:
\[ \Rightarrow x > 11................................\left( v \right)\]
Therefore on solving $3x + 2 < 5\left( {x - 4} \right)$ the range of the unknown value $x$ would be \[x > 11\].It means that the inequality $3x + 2 < 5\left( {x - 4} \right)$ would be satisfied for all the values of $x$ which are greater than the number $11$.

Note:Inequalities are mathematically expressed expressions which contain the symbols $ < ,\, > ,\, \leqslant ,\, \geqslant $. Inequalities can be solved by using algebra as well as by using graphs. The above question is solved algebraically. Also solving an inequality simply means to find a range of the unknown variable which would satisfy the inequality.