Chapter 6.5, Problem 24HP

To determine

To calculate: A linear inequality equation which pass through the given points.

Answer to Problem 24HP

The equation of inequality is $y-3x\ge -2$ .

Explanation of Solution

Given information: The inequality has $\left(-4,-14\right),\left(4,10\right)\&\left(8,22\right)$ as its solutions.

Formula used:

Two-point form of line is

  $\left(y-y_1\right)=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\times \left(x-x_1\right),\text{where}\left(x_1,y_1\right)\text{and}\left(x_2,y_2\right)\text{are}\text{the}\text{two}\text{points}\text{on}\text{the}\text{line}$.

Calculation:

Since the inequality has $\left(-4,-14\right),\left(4,10\right)\&\left(8,22\right)$ as its solutions then the line passing these 3 points can be written by taking the first two points $a=\left(-4,-14\right)andb=\left(4,10\right)$ and the equation of the line is

  $\begin{array}{l}\left(y-y_1\right)=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\left(x-x_1\right)\\ \left(y-\left(-14\right)\right)=\frac{\left(10-\left(-14\right)\right)}{\left(4-\left(-4\right)\right)}\left(x-\left(-4\right)\right)\\ \left(y+14\right)=3\left(x+4\right)\\ y-3x=-2\end{array}$

Hence the equation of line passing from these 3 points is $y-3x=-2$ .

Now the inequality equation can be $y-3x\le -2ory-3x\ge -2$ , Since it is not mention in the question. Hence any one of them is correct.

Therefore the equation of inequality is $y-3x\ge -2$ .