Question

How do you solve for x in $3x - 5 < x + 9 \leqslant 5x + 13$ ?

Answer 1

Hint:In this question, we are given that $3x - 5 < x + 9 \leqslant 5x + 13$ , where “<” indicates that the term on the right side is greater than term on the left side and indicates that the term on the right side is greater than or equal to the term on the left. So, we have \[x + 9\] is greater than $3x - 5$ and smaller than or equal to $5x + 13$ , it is an algebraic expression containing one unknown variable quantity (x). We know that we need an “n” number of equations to find the value of “n” unknown variables. In the given algebraic expression; we have exactly one equation and 1 unknown quantity, so by applying the given arithmetic operations, we can find the value of x.

Complete step by step answer:
We are given that $3x - 5 < x + 9 \leqslant 5x + 13$
We can write the above equation as –
$3x - 5 < x + 9$ and $x + 9 \leqslant 5x + 13$
To find the value of x, we will rearrange the equations as follows –
$ \Rightarrow 3x - x < 9 + 5$ and $9 - 13 \leqslant 5x - x$
$ \Rightarrow 2x < 14$ and $ - 4 \leqslant 4x$
$ \Rightarrow x < 7$ and $ - 1 \leqslant x$
We combine the above two inequalities and get –
$ - 1 \leqslant x < 7$
Hence, when $3x - 5 < x + 9 \leqslant 5x + 13$ , we get the value of x in the interval $[ - 1,7)$ .

Note: In the given algebraic expression, the alphabet representing the unknown quantity has a non- negative integer as power, that is, 1. So the given expression is a polynomial equation and is known as a linear equation as the degree of x is 1 (degree is defined as the highest power of the unknown quantity in a polynomial equation). We can easily solve similar equations by rearranging the equation such that one side of the equation contains the terms containing x and all other terms lie on the other side.