Question

Solve the inequality \[7 \leqslant \dfrac{{3x + 11}}{2} \leqslant 11\]

Answer 1

Hint: We have to find the value of \[x\] from the given expression of inequality \[7 \leqslant \dfrac{{3x + 11}}{2} \leqslant 11\] . We solve this question using the concept of solving linear equations of inequality . First we would simplify the terms of the both sides by cross multiplying both sides by \[2\] , we would obtain an inequality relation in terms of \[x\] . On further solving the expression we will get the range for the value of \[x\] for which it satisfies the given expression .

Complete step-by-step solution:
Given :
\[7 \leqslant \dfrac{{3x + 11}}{2} \leqslant 11\]
Cross multiply both sides of the expression of inequality by \[2\] , we get
\[7 \times 2 \leqslant \left( {3x + 11} \right) \leqslant 11 \times 2\]
\[14 \leqslant 3x + 11 \leqslant 22\]
Now , we will simplify the terms of the inequality in terms of \[x\] only such that we will obtain an inequality for the range of \[x\] which will satisfy the given expression .
Simplifying the terms of inequality , we get
Subtracting \[11\] from both sides , we get
\[14 - 11 \leqslant 3x \leqslant 22 - 11\]
\[3 \leqslant 3x \leqslant 11\]
On further solving the expression of inequality , we get
Dividing both sides of the inequality by \[3\] , we get
\[\dfrac{3}{3} \leqslant x \leqslant \dfrac{{11}}{3}\]
\[1 \leqslant x \leqslant \dfrac{{11}}{3}\]
Hence , the solution of the inequality \[7 \leqslant \dfrac{{3x + 11}}{2} \leqslant 11\] is \[\left[ {1,\dfrac{{11}}{3}} \right]\].

Note: We must take care about the sign and symbols of the inequality , as a slight change causes major errors in the solution . The solution of the range of the inequality states that each and every value which lies in that particular range satisfies the given equation . The square bracket \[\left[ {} \right]\] in the value of the range states that the end elements i.e. \[1\] and \[\dfrac{{11}}{3}\] in this question will also satisfy the given expression whereas the round bracket \[\left( {} \right)\] states that the end elements of the range will not satisfy the given expression .