Chapter 5.7, Problem 45E

(a)

To determine

To write: a system of linear inequalities that has no solution.

Answer to Problem 45E

The possible answer is $y<2x-1$ .

Explanation of Solution

Given information:

One inequality in a system is

  $-4x+2y>6$

Calculation:

Rewrite the inequality in slope intercept form as shown below

  $-4x+2y>6$

Add $4x$ from both sides of equation

  $\begin{array}{c}4x-4x+2y>6+4x\\ 2y>6+4x\end{array}$

Divide 2 from both sides of equation.

  $\begin{array}{c}\frac{2y}{2}>\frac{6}{2}+\frac{4x}{2}\\ y>3+2x\end{array}$

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 equation should have same slope and it solution lies below the line.

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

(b)

To determine

To write: a system of linear inequalities that has infinitely many solutions.

Answer to Problem 45E

The possible answer is $y>x$ .

Explanation of Solution

Given information:

One inequality in a system is

  $-4x+2y>6$

Calculation:

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

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