We are working on inequalities now in math, and simplifying them and whatnot.
I cant quite get these:
1<2x-3<5
-7<3-1/4x<1
2x<-10 or x + 3>1
Please show me how you got it!
I cant quite get these:
1<2x-3<5
-7<3-1/4x<1
2x<-10 or x + 3>1
Please show me how you got it!
-
So the trick in manipulating inequalities or equations is just to do the same to every value. You just have to remember that multiplying by a negative number flips the sign of inequalities.
1 < 2x-3 < 5
1+3 < 2x-3+3 < 5+3
4 < 2x < 8
4(1/2) < 2x(1/2) < 8 note how I'm multiplying by 1/2 instead of dividing; that's a good way to remember that division can cause signs to flip too. It doesn't here, though, because 1/2 is not negative.
2 < x < 4.
-7 < 3-(1/4)x < 1
-7-3 < 3-3-(1/4)x < 1-3
-10 < -(1/4)x < -2
-10(-4) > -(1/4)(-4)x > -2(-4) note I'm flipping both signs -- I multiplied by -4, which is negative.
40 > x > 8
For the third one, you've got two separate inequalities, we just have to get x alone in each one.
2x < -10
x < -5
or
x+3 > 1
x > 1-3 = -2
so x < -5 or x > -2.
For the first two, we had inequalities limiting x to some range; there was a lower and upper bound for each. In this case, we have inequalities limiting x to *outside* the range -5 to -2. It's important to understand that in case one of the inequalities turns out to be redundant, or in case two inequalities are impossible to meet at the same time, in which case they can be simplified.
1 < 2x-3 < 5
1+3 < 2x-3+3 < 5+3
4 < 2x < 8
4(1/2) < 2x(1/2) < 8 note how I'm multiplying by 1/2 instead of dividing; that's a good way to remember that division can cause signs to flip too. It doesn't here, though, because 1/2 is not negative.
2 < x < 4.
-7 < 3-(1/4)x < 1
-7-3 < 3-3-(1/4)x < 1-3
-10 < -(1/4)x < -2
-10(-4) > -(1/4)(-4)x > -2(-4) note I'm flipping both signs -- I multiplied by -4, which is negative.
40 > x > 8
For the third one, you've got two separate inequalities, we just have to get x alone in each one.
2x < -10
x < -5
or
x+3 > 1
x > 1-3 = -2
so x < -5 or x > -2.
For the first two, we had inequalities limiting x to some range; there was a lower and upper bound for each. In this case, we have inequalities limiting x to *outside* the range -5 to -2. It's important to understand that in case one of the inequalities turns out to be redundant, or in case two inequalities are impossible to meet at the same time, in which case they can be simplified.