Chapter 2, Problem 2CA

(a)

To determine

To find: the value of b .

Answer to Problem 2CA

  $b=-2$

Explanation of Solution

Given information: an inequality is given as follows, $ax+4\le 3x+b$

And we have to find the value of $b$ that satisfies

  $\text{When }a=\text{5 and }b=\text{\_\_\_\_\_\_, }x\le -3$

Putting $a=5$ in the given inequality $ax+4\le 3x+b$ we get

  $\begin{array}{l}5x+4\le 3x+b\\ \Rightarrow 5x-3x\le b-4\\ \Rightarrow 2x\le b-4\\ \Rightarrow x\le \frac{b-4}{2}\end{array}$

And comparing with $x\le -3$

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

(b)

To determine

To find: the values of a and b.

Answer to Problem 2CA

  $a=3\text{ and }b=5$

Explanation of Solution

Given information: an inequality is given as follows, $ax+4\le 3x+b$

And we have to find the values of $a\text{ and }b$ that satisfies

  $\text{When }a=\_\_\_\_\text{ and }b=\text{\_\_\_\_\_\_, the solution of inequality is all real numbers}$

Putting $a=3\text{ and }b=5$ in the given inequality $ax+4\le 3x+b$ we get

  $\begin{array}{l}3x+4\le 3x+5\\ \Rightarrow 4\le 5\end{array}$

This is possible for all real values of x.

(c)

To determine

To find: the values of a and b.

Answer to Problem 2CA

  $a=3\text{ and }b=3$

Explanation of Solution

Given information: an inequality is given as follows, $ax+4\le 3x+b$

And we have to find the values of $a\text{ and }b$ that satisfies

  $\text{When }a=\_\_\_\_\text{ and }b=\text{\_\_\_\_\_\_, the inequality has no solution}\text{.}$

Putting $a=3\text{ and }b=3$ in the given inequality $ax+4\le 3x+b$ we get

  $\begin{array}{l}3x+4\le 3x+3\\ \Rightarrow 4\le 3\end{array}$

This is not possible, hence there is no solution of