Chapter 12, Problem 60CUR

To determine

To solve : The inequality, graph and check the solution.

Answer to Problem 60CUR

The solution is $x\ge \frac{17}{6}$ . i.e. all the numbers greater than or equal to $\frac{17}{6}$ is the solution.

Explanation of Solution

Given information : $-\frac{2}{7}x-\frac{2}{3}\le \frac{1}{7}$

Formula used : Adding or subtracting the same number on each side of an inequality produces an equivalent inequality.

Multiplying each side of an inequality by a positive number produces an equivalent inequality.

Multiplying each side of an inequality by a negative numbers and reversing the direction of the inequality symbol produces an equivalent inequality.

Dividing each side of an inequality by a positive number produces an equivalent inequality.

Dividing each side of an inequality by a negative numbers and reversing the direction of the inequality symbol produces an equivalent inequality.

Calculation :

L.C.M of $7$ and $3$ is $21$

  $\begin{array}{l}-\frac{2}{7}x-\frac{2}{3}\le \frac{1}{7}\\ 21\times \left(-\frac{2}{7}x-\frac{2}{3}\right)\le 21\times \frac{1}{7}\\ 21\times \left(-\frac{2}{7}\right)x-21\times \frac{2}{3}\le 3\times 1\\ -\left(3\times 2\right)x-7\times 2\le 3\\ -6x-14\le 3\\ -6x-14+14\le 3+14\\ -6x\le 17\\ \frac{-6x}{-6}\le \frac{17}{6}\\ x\ge \frac{17}{6}\end{array}$

Check:

$7$ is the number greater than or equal to $\frac{17}{6}=2\frac{5}{6}$ . Putting $7$ on the place of $x$ in the left hand side of the given inequality we get,

  $\begin{array}{c}-\frac{2}{7}x-\frac{2}{3}=-\frac{2}{7}\times \left(7\right)-\frac{2}{3}\\ =-2-\frac{2}{3}\\ =\frac{-\left(2\times 3\right)-2}{3}\\ =\frac{-6-2}{3}\\ =-\frac{8}{3}\end{array}$

  $-\frac{8}{3}\le \frac{1}{7}$

Therefore, the solution is right.

Graph :

The solution is $x\ge \frac{17}{6}$ .

Interpretation : The solution is $x\ge \frac{17}{6}$ . i.e. all the numbers greater than or equal to $\frac{17}{6}$ is the solution.