Given- The set of inequalities,
[tex]3x+43x-10}[/tex][tex]-4x-3\leq2x+9\text{ and -3x+2>-2x+5}[/tex][tex]4(x+1)\ge3(x+2)\text{ and -3\lparen x-1\rparen<5\lparen x+2\rparen}[/tex][tex]4x-6\leq5x+6\text{ and -3x-2>2x+8}[/tex]
Required- To find out which one of these inequalities have no solution.
Explanation- The inequality which does not give any possible value of the variable will result in no solution.
Now, consider the first inequality,
Solving the inequality further,
[tex]3x+4\lt x-8\text{ and -5x+4\gt3x-10}[/tex][tex]3x-x<-8-4\text{ and -5x-3x>-10-4}[/tex][tex]2x<-12\text{ and -8x>-14}[/tex][tex]which\text{ gives us , }x<-6\text{ and }x<\frac{14}{8}[/tex]
which will give us many solutions such as -7,-8,-9,
and so on.
The values of x that are less than 14/8 and -6 exists.
Remember to change the sign of inequality while multiplying it by negative number.
Now similarly if we solve our second inequality for x we get,
[tex]x\ge-2\text{ and }x<-3[/tex]
This is the inequality that will result in no solution. Since,
[tex]\begin{gathered} x\ge-2\text{ gives , }x=\text{ -2,-1,0,1 etc and} \\ x<-3\text{ gives, }x=-4,-5,-6\text{ and so on } \end{gathered}[/tex]
As there is no possible value of x that satisfies both our inequality. Hence equality 2 will result in no solution.
Final Answer- Option B
