Question 42794
a) x+3>5x-21
-4x+3>-21
-4x>-24
x<(-24)/(-4)
x<6


b) |1+2x|>=5
this "absolute" question forms a V-shape if we plot it. The equations of the 2 parts of the V are 1+2x and -(1+2x) so we need to solve both parts ie ask where are both lines greater or equal to 5.


Right then:
{{{ 1+2x>=5 }}}
{{{ 2x >= 4 }}}
{{{ x >= 4/2 }}}
{{{ x >= 2 }}}


and:
{{{ -(1+2x)>=5 }}}
{{{ -1-2x>=5 }}}
{{{ -2x>=6 }}}
{{{ x<=6/(-2) }}}
{{{ x<=-3 }}}


so, we have x less than or equal to -3
or we have x is greater than or equal to 2


c) x^2-x-6>=0
first do x^2-x-6=0
(x-3)(x+2)=0
so x-3=0 or x+2=0
so x=3 or x=-2.


Now, you need to visualise this curve. It is a u-shape cutting the x-axis at x=-2 and x=3.


The question asked, where is it greater or equal to zero? The answer is everywhere on the curve above the x-axis. So the answer is values of x less than or equal to -2 and also values of x greater than or equal to 3.


d) x-1/2-x >= 0

This is a tricky one... you need to know that this form of equation has 2 distinct parts, separateed by asymptotes. We need to find the asymptotes and where the curve(s) cross the axis. Then we can sketch it and then we can answer the question.


{{{ (x-1)/(2-x) = 0 }}}
asymptote is when 2-x=0 --> x=2


Now for the y-asymptote:
{{{ y = (x-1)/(2-x) }}}
{{{ y(2-x) = (x-1) }}}
{{{ 2y-xy = x-1 }}}
{{{ 2y+1 = x+xy }}}
{{{ 2y+1 = x(1+y) }}}
{{{ x = (2y+1)/(1+y) }}}
asymptote is when 1+y=0 --> y=-1.


Now for where the curves cross the axes:
when x=0, {{{ y = (x-1)/(2-x) }}}
--> {{{ y = (0-1)/(2-0) }}}
--> {{{ y = (-1)/(2) }}}
--> {{{ y = (-1)/2 }}}


when y=0, {{{ x = (2y+1)/(1+y) }}}
--> {{{ x = (2(0)+1)/(1+0) }}}
--> {{{ x = (1)/(1) }}}
--> x=1


So draw axes.
Draw 2 asymptotes, at x=2 and y=-1
Plot 2 point: (-1/2, 0) and (0,1)
Sketch one part of the curve through these 2 points approaching the asymptotes. The other half of the curve is in the diagonally opposite quadrant.


Now to answer the question:
where is the curve greater or equal to zero. By looking at the sketch, the curve is greater or equal to zero from x>=1 upto x<2.


e) 3x+1/2plus 4 < x/2
not sure what you mean here...you try



f) |1-3x|<5
same approach as other || question. You try it.


jon.