Chapter 1.5, Problem 48HP

To determine

To write: an inequality for which the solution is all real numbers in the form $ax+b>c\left(x+d\right)$ .

Answer to Problem 48HP

The answer is

  $a=c$ and $b>c\cdot d$ .

Explanation of Solution

For inequality to have solution in $x\in ℝ$ , to find a way to not have any $x$ on each side of inequality and apart from that, the rest of the numbers have to make true statement. So let’s begin by expanding brackets.

  $ax+b>cx+cd$

From our first logical statement, where need to not have any $x$ in inequality, it is easy to determine that $a=c$ and that it is only way for $x$ to cancel out on each side,

Next step, after cancelling $x$ will left us:

  $b>cd$

Which means that $b$ has to be number greater than $c\cdot d$ , no matter what the $c$ is, as it is equal to $a$ .

For example if $a=c=4,b=9$ and $d=2$ which means that $b>cd$ or $9>4\cdot 2$ what is correct by our statements, we will have:

  $4x+9>4x+8$

Which is true, because after cancel $4x$ , we are left with $9>8$ .

So, the answer is

  $a=c$ and $b>c\cdot d$ .