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# Solve the inequalities and represent the solution graphically on number line. $2(x-1) < x+5 ; 3(x+2) > 2-x$

$\begin{array}{1 1}(A)\;(-1,7) \\(B)\;(-5,5)\\(C)\;(-23,2]\\(D)\;[2,3]\end{array}$

Toolbox:
• Same Quantity can be added (a subtracted ) to (from ) both sides of the inequality with out changing the sign of the in equality.
• Same positive quantities can be multiplied or divided to both side of the in equality with out changing the sign of the inequality.
• If same negative quantity is multiplied or divided to both sides of the inequality is reversed i.e $'>'$ sign changes to $'<'$ and $'<'$ changes $'>'$ .
• To represent solution of linear inequality involving one variable on a number line, if the inequality involves $\geq$ or $\leq$ are draw filled circle (0) on the number is included in the solution set.
• If the inequality involves $'>'$ or $'<'$ we draw open circle (0) on the number line to indicate the number is excluded from the solution set.
Step 1:
The first inequality is $2(x-1) < x+5$
$=> 2x-2 < x+5$
Subtracting x from both sides of inequality and adding 2 on both sides of inequality
$=> 2x -x -2+2 < x -x +5+2$
$=> x < 7$-----(1)
Step 2:
The second inequality is $3(x+2) > 2-x$
$=> 3x+6 > 2-x$
Adding x on both sides of the inequality and subtracting 6 from both sides of inequality .
$=> 3x+6 +x-6 > 2-x +x-6$
$=> 4x> -4$
Dividing by 4 on both sides of inequalities $=> x >-1$-----(2)
Step 3:
From (1) and (2) we conclude that all numbers greater than -1 and less than T are solutions to the given inequality
The solution set is $(-1,7)$
The solution is represented graphically as the number line as
edited Aug 5, 2014 by meena.p