Question:
Solution:
Consider the following inequalities system :
Inequality 1:
[tex]-6(x-2)\text{ }\leq36[/tex]
or
Inequality 2:
[tex]4+x<14[/tex]
Applying the distributive property in inequality 1, we obtain:
[tex]-6x+12\text{ }\leq36[/tex]
this is equivalent to:
[tex]-6x\text{ }\leq36-12\text{ = 24}[/tex]
that is:
[tex]-6x\leq24[/tex]
this is equivalent to:
[tex]6x\ge-24[/tex]
solving for x, we get:
[tex]x\text{ }\ge-\frac{24}{6}\text{ = -4}[/tex]
that is:
[tex]x\text{ }\ge\text{ -4}[/tex]
Then inequality 1 is equivalent to the following solution
[tex]x\text{ }\ge\text{ -4}[/tex]
On the other hand, for inequality 2 solving for x, we get:
[tex]x<14-4\text{ = 10}[/tex]
that is:
[tex]x<10[/tex]
so that, the solution to the inequality system is
[tex]x\text{ }\ge\text{ -4}[/tex]
or
[tex]x<10[/tex]
now, this is equivalent to say:
[tex]x\text{ }\ge\text{ -4 U x<10}[/tex]
or in interval notation:
[tex]\lbrack-4,+\infty)\text{ U (-}\infty\text{,10) }[/tex]
