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Question 706638: 1. 28 +14x greater than or equal to 7 AND 8x-3<33 - Need notations
2. x+10<2x-3 AND 6x+2>9x-2 - Need notations
3. 7x-3<3x+11 AND 2x-7 less than or equal to 3x+5 - need notations
4. 3x+12 < 6x+6 AND x-8 greater than or equal to 4x+1 - Need notations
5. 3-2x>7 AND 5x-2 greater than -18 - Need notations
Answer by DrBeeee(684)   (Show Source):
You can put this solution on YOUR website! I'll show you how to do one, then you should be able to do the others.
The intersection or AND term only means that the variable will occur between two numbers. To solve, you solve each of the two inequalities separately. Let's do the first one.
Solve the left inequality
(1) 28 + 14*x >= 7
Let's simplify by dividing through by 7 and get
(2) 4 + 2*x >= 1, where we do not reverse the inequality because we divided by a positive number, +7.
Now solve (2) for x
(3) 2*x >= 1 - 4 or
(4) 2*x >= -3 or
(5) x >= -1.5 , where again the inequality remains the same because we divided both side by +2.
Now let's do the second part of problem 1
(6) 8*x - 3 < 33
Solve (6) for x
(7) 8*x < 33 + 3 or
(8) 8*x < 36 or
(9) x < 36/8 or
(10) x < 4.5, and again the inequality stays the same.
Now do the AND by writing the compound (two sided) inequality
(11) - 1.5 <= x < 4.5
Note that the permissible set (domain) of values for x lies on -1.5 and BETWEEN -1.5 AND +4.5, but not on 4.5.
Do the others the same way.
I reviewed my answer and the other questions. I better help you with 5) because it has a negative coefficient for x, so you must be careful. The books tell you to multiply by a negative number which requires a reversal of the inequality, but I'll show you how to avoid that.
The first part of 5) is
(12) 3 - 2*x > 7 or
(13) -2*x > 7 - 3 or
(14) -2*x > 4
Now text books tell you to divide by (-2) and get
(15) x > 4/(-2) or
(16) x > -2, however if you forget the reversal rule and use (16) as your answer - you'd be wrong! When you multiply or divide by a negative number you must reverse the inequality. Since we divided by (-2), (16) should be changed to
(17) x < -2 as the correct inequality.
How does DrBeee avoid this? Look at (14) and do the following,
Add 2*x to both sides and get
(18) -2*x + 2*x > 4 + 2*x or
(19) 0 > 4 + 2*x
Now subtract 4 from each side and get
(20) -4 > 2*x or
by dividing by +2 we get
(21) x < -2.
My approach may seem longer but I avoid making mistakes. Please feel free to take your choice.
Now do the second part of problem 5) which is
(22) 5*x - 2 > -18 or
(23) 5*x > -16 or
(24) x > -3.2
The answer to problem 5 is
(25) -3.2 < x < -2
In this case, the solution domain of x lies between -3.2 AND -2.
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