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It is so called compound inequality
6x + 4 < 2 OR 6x + 4 > 10.
Notice that the two inequalities are connected by the service word " OR ".
It means that the solution is THE UNION of the solution sets to each of the two participating inequalities.
For the first of the two inequalities the solution is x < = .
For the second inequality the solution is x > = 1.
So, the solution to the given compound inequality is the union of two sets
{ x | x < } U { x | x > 1}.
In interval form, it is the union of two intervals (,) U (1,oo).
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Comment from student : thank you so much. i have one more question
Triple a number, x, increased by 1 is between -20 and 10. What are the solutions for x?
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My response : In Math, using words instead of mathematical symbols, often leads to ambiguity.
In your case, these words " is between -20 and 10 " leave it unclear if the endpoints are included.
Therefore, I will use the sign " <= " for certainty.
-20 <= 3x + 1 <= 10
Subtract 1 (one) from both sides. You will get an equivalent inequality
-20 - 1 <= 3x <= 10 - 1, or
-21 <= 3x < 9.
Now divide all three terms of the last inequality by 3. You will get
-7 <= x <= 3.
It is your ANSWER : -7 <= x <= 3.
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For inequalities, see introductory lessons
- Solving simple and simplest linear inequalities
- Solving systems of linear inequalities in one unknown
- Solving compound inequalities
in this site.
Consider them as your textbook, handbook, guide, tutorial and (free of charge) home teacher.
Learn the subject from there once and for all.
Happy learning !