Question

2 ) solve and graph the inequality
$ - \frac { 1 } { 3 } x < - \frac { 3 } { 4 } x - 5 + \frac { 5 } { 2 } x $

Explanation:

Step1: Combine like terms

Combine the \(x\) - terms on the right - hand side: \(-\frac{3}{4}x+\frac{5}{2}x=-\frac{3}{4}x+\frac{10}{4}x=\frac{-3 + 10}{4}x=\frac{7}{4}x\).
The inequality becomes \(-\frac{1}{3}x<\frac{7}{4}x-5\).

Step2: Move \(x\) - terms to one side

Add \(\frac{1}{3}x\) to both sides: \(-\frac{1}{3}x+\frac{1}{3}x<\frac{7}{4}x+\frac{1}{3}x - 5\).
\(0<\frac{21x + 4x}{12}-5\), so \(0<\frac{25}{12}x-5\).

Step3: Isolate \(x\)

Add \(5\) to both sides: \(5<\frac{25}{12}x\).
Multiply both sides by \(\frac{12}{25}\): \(5\times\frac{12}{25}\(\frac{12}{5} \frac{12}{5}=2.4\).

Answer:

The solution of the inequality is \(x>2.4\). On the number - line, we draw an open circle at \(x = 2.4\) (since the inequality is strict, \(x
eq2.4\)) and shade the region to the right of \(2.4\).