Chapter 5, Problem 15MCQ

To determine

To solve the inequality:

  $-9a\ge -45$

And check the solution

Answer to Problem 15MCQ

  $\left(-\infty ,5\right)$

Explanation of Solution

Given:

  $-9a\ge -45$

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. $b\le c\Rightarrow ab\le ac,\text{ if }a\ge 0$
2. $b\le c\Rightarrow ab\ge ac,\text{ if }a\le 0$
3. $a\le b\Rightarrow -a\ge -b$

Calculation:

To solve the inequality:

  $-9a\ge -45$

Divide both sides of the inequality with $9$ , as

  $\begin{array}{l}-9a\ge -45\\ \Rightarrow \frac{-9a}{9}\ge \frac{-45}{9}\\ \Rightarrow -a\ge -5\\ \Rightarrow a\le 5\end{array}$

Thus, the inequality is true for a less than or equal to $5$ , thus the solution of given inequality in interval form is:

  $\left(-\infty ,5\right)$

Now, to check the answer solution for the given inequality first check the solution at end point by converting it into an equation. Putting a equals to $5$ in left hand side

  $\begin{array}{c}-9\left(5\right)=-45\\ =\text{Right Hand}\end{array}$

Now put any value any value of a less than $5$ , say $2$ and check whether it satisfies the inequality, as

  $\begin{array}{l}-9\left(2\right)=-18\\ \text{And}-18>-45\end{array}$

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