Question:
Solve and graph the compound inequality on the real numbers and write the solutions in interval notation, and provide one specific solution.
3x/4-1>x/3-3/5 and 2x-18/5>x+2/5.
Solution:
Consider the following inequality:
[tex]\frac{3x}{4}-1\text{ > }\frac{x}{3}-\frac{3}{5}[/tex]Putting the similar terms together, we get:
[tex]\frac{3x}{4}-\text{ }\frac{x}{3}\text{>}-\frac{3}{5}+1[/tex]this is equivalent to:
[tex]\frac{9x-4x}{12}>\frac{2}{5}[/tex]this is equivalent to:
[tex]\frac{5x}{12}\text{ >}\frac{2}{5}[/tex]this is equivalent to:
[tex]25x\text{ > 24}[/tex]this is equivalent to:
[tex]x>\text{ }\frac{24}{25}\text{ = 0.96}[/tex]in interval notation, this is equivalent to:
[tex](0.96,\text{ }\infty)[/tex]and the graph is:
Now, consider the following inequality:
[tex]2x-\frac{18}{5}\text{ > }x+\frac{2}{5}[/tex]Putting the similar terms together, we get:
[tex]2x-x\text{ > }\frac{2}{5}+\frac{18}{5}[/tex]this is equivalent to:
[tex]x\text{ > }\frac{20}{5}=\text{ 4}[/tex]in interval notation, this is equivalent to:
[tex](4,\infty)[/tex]and the graph is:
