first thing is you want to find the limits of x within the square root sign.
square root of a function requires the function to be >= 0 in order to get an answer that is real, rather than complex or imainary.
start with 8 - 5x >= 0
add 5x to both sides to get 8 >= 5x
divide both sides by 5 to get 8/5 >= x
solve for x to get x <= 8/5
start with x + 7 >= 0
subtract 7 from both sides to get x >= -7
your limits for x within the square root sign are:
x >= -7
x <= 8/5
back to your inequality.
start with sqrt(8-5x) <= sqrt(x+7)
square both sides to get 8-5x <= x+7
add 5x to both sides and subtract 7 from both sides to get 1 <= 6x
divide both sides by 6 to get 1/6 <= x
solve for x to get x >= 1/6
your 3 inequalities that have to be satisfied are:
x >= -7
x <= 8/5
x >= 1/6
since 1/6 is already greater than -7, your inequalities that have to be satisfied are:
x <= 8/5
x >= 1/6
your solution therefore becomes 1/6 <= x <= 8/5
that is the region where you will get a real answer and satisfy the original inequality.
you can also find your solution by graphing.
you need to graph:
y = sqrt(8-5x) and y = sqrt(x + 7)
you then find the region on the graph where sqrt(8-5x) is <= sqrt(x+7)
the graph is shown below:
note that 1/6 = .167 rounded to 3 decimal places and 8/5 = 1.6 as shown for the values of x in the coordinate points represented in (x,y) format.
