Question
Solve the following system of linear inequalities $\frac{4 x}{3}-\frac{9}{4}x$
✓
Answer
We have, $\frac{4 x}{3}-\frac{9}{4} $ and $\frac{7 x-1}{3}-\frac{7 x+2}{6}>x \ldots(ii)$
From inequality $(i),$
we get $\frac{4 x}{3}-\frac{9}{4}$
$\Rightarrow 16 x-27<12 x+9 \ [$multiplying both sides by $12]$
$\Rightarrow 16 x-27+27<12 x+9+27 \ [$adding $ 27 $ on both sides$]$
$\Rightarrow 16 x<12 x+36$
$\Rightarrow 16 x-12 x<12 x+36-12 x \ [ $subtracting $ 12 x$ from bot sides$]$
$\Rightarrow 4 x<36 $
$\Rightarrow x<9 \ [$dividing both sides by $4 ]$
Thus, any value of $x$ less than $9$ satisfies the inequality.
So, the solution of inequality $(i)$ is given by $x \in(-\infty, 9)$
From inequality $(ii)$ we get,
$\frac{7 x-1}{3}-\frac{7 x+2}{6}>x $
$\Rightarrow \frac{14 x-2-7 x-2}{6}>x$
$\Rightarrow 7 x-4>6 x \ [$multiplying by $6 $ on both sides$]$
$\Rightarrow 7 x-4+4>6 x+4 \ [$adding $ 4 $ on both sides$]$
$\Rightarrow 7 x>6 x+4$
$\Rightarrow 7 x-6 x>6 x+4-6 x \ [$subtracting $ 6 x $ from both sides$]$
$\therefore x>4$
Thus, any value of $x$ greater than $4$ satisfies the inequality.
So, the solution set is $x \in(4, \infty)$
The solution set of inequalities $(i)$ and $(ii)$ are represented graphically on number line as given below:
Clearly, the common value of $x$ lie between $4$ and $9$
Hence, the solution of the given system is, $4$
From inequality $(i),$
we get $\frac{4 x}{3}-\frac{9}{4}$
$\Rightarrow 16 x-27<12 x+9 \ [$multiplying both sides by $12]$
$\Rightarrow 16 x-27+27<12 x+9+27 \ [$adding $ 27 $ on both sides$]$
$\Rightarrow 16 x<12 x+36$
$\Rightarrow 16 x-12 x<12 x+36-12 x \ [ $subtracting $ 12 x$ from bot sides$]$
$\Rightarrow 4 x<36 $
$\Rightarrow x<9 \ [$dividing both sides by $4 ]$
Thus, any value of $x$ less than $9$ satisfies the inequality.
So, the solution of inequality $(i)$ is given by $x \in(-\infty, 9)$
From inequality $(ii)$ we get,
$\frac{7 x-1}{3}-\frac{7 x+2}{6}>x $
$\Rightarrow \frac{14 x-2-7 x-2}{6}>x$
$\Rightarrow 7 x-4>6 x \ [$multiplying by $6 $ on both sides$]$
$\Rightarrow 7 x-4+4>6 x+4 \ [$adding $ 4 $ on both sides$]$
$\Rightarrow 7 x>6 x+4$
$\Rightarrow 7 x-6 x>6 x+4-6 x \ [$subtracting $ 6 x $ from both sides$]$
$\therefore x>4$
Thus, any value of $x$ greater than $4$ satisfies the inequality.
So, the solution set is $x \in(4, \infty)$
The solution set of inequalities $(i)$ and $(ii)$ are represented graphically on number line as given below:
Clearly, the common value of $x$ lie between $4$ and $9$
Hence, the solution of the given system is, $4$
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