6x2+44x+70=(2x+10)(3x+7)≥0
The product of two values is positive only if both values are positive or both values are negative. Thus
2x+10≥0and3x+7≥0
or
2x+10≤0and3x−7≤0
Solving for x in each inequality, we find
x≥−5andx≥−73
or
x≤−5andx≤−73
Because −73>−5, we have that x≥−73 implies x≥−5 already. Similarly, x≤−5 implies x≤−73. Thus, we can rewrite the pairs of inequalities as single inequalities:
x≥−73orx≤−5
Using interval notation, we can express the set of values which act as solutions for each inequality (note that ∈ means "is an element of" or "is in":)
x≥−73⇔x∈[−73,∞)
(x is in the interval containing all real numbers from and including −73 to infinity)
x≤−5⇔x∈(−∞,−5]
(x is in the interval containing all real numbers from negative infinity to, and including, −5)
The ∪, or "union" symbol allows us to treat multiple sets as a single set. If A and B are sets, then A∪B is the set which contains all the elements of A and all the elements of B. Thus, we can write our entire solution set as a single set by using ∪ to combine the intervals.
x∈(−∞,−5]∪[−73,∞)
