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

Last updated date: 18th Jun 2024
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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.

We are given that $3x - 5 < x + 9 \leqslant 5x + 13$
$3x - 5 < x + 9$ and $x + 9 \leqslant 5x + 13$
$\Rightarrow 3x - x < 9 + 5$ and $9 - 13 \leqslant 5x - x$
$\Rightarrow 2x < 14$ and $- 4 \leqslant 4x$
$\Rightarrow x < 7$ and $- 1 \leqslant x$
$- 1 \leqslant x < 7$
Hence, when $3x - 5 < x + 9 \leqslant 5x + 13$ , we get the value of x in the interval $[ - 1,7)$ .