Chapter 2, Problem 2CA

(a)

To determine

To fill: The missing values so that each statement is true

Answer to Problem 2CA

The missing value is $b=-2$

Explanation of Solution

Given:

  $ax+4\le 3x+b$

  $a=5$

Calculation:

The inequality is given

  $ax+4\le 3x+b$

Substitute $a=5$

  $5x+4\le 3x+b$

  $5x+4-3x\le 3x+b-3x$

  $2x+4\le b$

  $2x+4-4\le b-4$

  $2x\le b-4$

  $x\le \frac{b-4}{2}$

To find $b$

  $\begin{array}{l}\frac{b-4}{2}=-3\\ b-4=-6\\ b=-2\end{array}$

Conclusion:

Hence, when $a=5$ the value of $b=-2$ $,x\le -3$

(b)

To determine

To fill in: The missing values so that each statement is true

Answer to Problem 2CA

The missing value is $b=-2$

  $a=3,b\ge 4$

Explanation of Solution

Given:

  $ax+4\le 3x+b$

Calculation:

The inequality is given

  $ax+4\le 3x+b$

Isolate $x$

  $\begin{array}{l}ax+4-3x\le 3x+b-3x\\ \left(a-3\right)x+4\le b\\ \left(a-3\right)x+4-4\le b-4\\ \left(a-3\right)x\le b-4\end{array}$

The Solution is all real numbers when we have

  $\begin{array}{l}a=3\\ b-4\ge 0\Rightarrow b\ge 4\end{array}$

Conclusion:

Hence, the Solution is all real numbers when we have $a=3,b\ge 4$

(c)

To determine

To fill in: The missing values so that each statement is true

Answer to Problem 2CA

There is no solution when we have $a=3,b<4$

Explanation of Solution

Given:

  $ax+4\le 3x+b$

Calculation:

The inequality is given

  $ax+4\le 3x+b$

There is no solution when we have

  $\begin{array}{l}a=3,b<4\\ b-4<0\Rightarrow b<4\end{array}$

Conclusion:

Hence, there is no solution when we have $a=3,b<4$