Chapter 5.7, Problem 45E

To determine

To find: another inequality so the system has no solution and infinitely many solutions.

Answer to Problem 45E

the possible answer is $y>x$ .

Explanation of Solution

Given:

  $-4x+2y>6$ .

Concept used:

The difference is that a solution to the inequality is not the drawn line but area of the coordinate plane that satisfy the inequality.

The common area is the solution of inequality.

The boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ .

Straight line equation:

  $y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$

Calculation:

Consider the system of linear inequality.

Step $1$ : rewrite the inequality in slope intercept form as shown below.

  $-4x+2y>6$ .

Add $4x$ from both the sides of equation.

  $\begin{array}{l}4x-4x+2y>6+4x.\\ 2y>6+4x.\\ y>3+2x.\end{array}$

This solution lies above the boundary line $y=2x+3$ .

In order to have no solution, the equation should be parallel and below the given line that is both equations should have same slope and the solution lies below the line.

Therefore, the possible answer is $y<2x-1$ .

Any linear inequality not parallel to the given will produce infinitely many solutions.

Hence, the possible answer is $y>x$ .