Chapter 1.7, Problem 28E

To determine

To calculate: The solution of the given inequality, $9x-1<\frac{3}{4}\left(16x-2\right)$, and graph the solution set.

The solution set of the given inequality, $9x-1<\frac{3}{4}\left(16x-2\right)$, is $x>\frac{1}{6}$.

Calculation:

Consider the given inequality, $9x-1<\frac{3}{4}\left(16x-2\right)$.

Multiply each part by 4 by using the multiplicative property of an inequality, according to which, if $c>0$, then $a<b$ becomes $ac<bc$ and if $c<0$, then $a>b$ becomes $ac<bc$.

$36x-4<48x-6$

Subtract $48x-4$ from each part by using the property of addition of a constant to an inequality, according to which, if $a<b$, then $a<b$ becomes $a+c<b+c$.

$-12x<-2$

Divide each part by $-12$ by using the multiplicative property of an inequality, according to which, if $c>0$, then $a<b$ becomes $ac<bc$ and if $c<0$, then $a>b$ becomes $ac<bc$.

$x>\frac{1}{6}$

The solution set of the given inequality is the set of all real numbers that are greater than $\frac{1}{6}$, denoted by $\left(\frac{1}{6},\infty \right)$.

Graph:

The solution set of the inequality is shown in the graph.

The parenthesis at $x=\frac{1}{6}$ means that the solution set does not include the value at $x=\frac{1}{6}$.