Chapter 5, Problem 17MCQ

To determine

To solve the inequality:

  $\frac{k}{7}<-2$

And check the solution

Answer to Problem 17MCQ

  $\left(-\infty ,-14\right)$

Explanation of Solution

Given:

  $\frac{k}{7}<-2$

Concept Used:.

• To check the answer solution for the given inequality

1. First check the solution at end point by converting it into an equation.
2. If the solution is $x<m$, then pick any value of x less than m and check whether it satisfies the inequality.
3. If the solution is $x>m$, then pick any value of x greater than m and check whether it satisfies the inequality
Properties of inequality:

1. $b<c\Rightarrow ab<ac,\text{ if }a>0$
2. $b<c\Rightarrow ab>ac,\text{ if }a<0$

Calculation:

To solve the inequality:

  $\frac{k}{7}<-2$

Multiply both sides of the inequality by

  $7$ , as

  $\begin{array}{l}\frac{k}{7}<-2\\ \Rightarrow 7\times \frac{k}{7}<7\left(-2\right)\\ \Rightarrow k<-14\end{array}$

Thus, the inequality is true for k strictly less than $-14$ , thus the solution of given inequality in interval form is:

  $\left(-\infty ,-14\right)$

Now, to check the answer solution for the given inequality first check the solution at end point by converting it into an equation. Putting k equals to $-14$ in left hand side

  $\begin{array}{c}\frac{-14}{7}=-2\\ =\text{Right Hand}\end{array}$

Now put any value any value of k less than $-14$ , say $-28$ and check whether it satisfies the inequality, as

  $\begin{array}{l}\frac{-28}{7}=-4\\ \text{And }-4<-2\end{array}$

Since, both conditions are satisfied thus solution of inequality is correct.