Question

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

Answer 1

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 .