How do you solve #1-9/8x >=8#?

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Answer 1 · 2017-03-07T12:54:47

#x<=-56/9#

There are two approaches to make #9/8x# positive

#color(blue)("Method 1")#

Add #9/8x# to both sides

#color(green)(1-9/8xcolor(red)(+9/8x)" ">=" "8color(red)(+9/8x))#

#" "1+0" ">=" "8+9/8x#

Subtract 8 from both sides

#" "1-8" ">= 0+9/8x#

Turn everything round so that #9/8x# is on the left

#" "9/8x <=-7" "..........Equation(1)#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Method 2")#

Multiply both sides by -1
This turns the inequality sign round the other way.

#" "-1+9/8x<=-8#

Add 1 to both sides

#" "9/8x<=-7" "...............Equation(2)#

Now compare equation(1) to equation(2). They are the same.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Multiply both sides by #color(red)(8/9)#

#" "color(green)(color(red)(8/9xx)9/8x" "<=" "(-7)color(red)(xx8/9))#

But #8/9xx9/8=1# giving

#" "x" "<=" "-56/9#