−7x+5≤−8x+6
Let's multiply both sides by −1. Since we are multiplying/dividing by a negative value, we must flip the direction of the inequality:
7x+5≥8x+6
Multiply both sides by (x+5):
7≥8(x+5)x+6
Multiply both sides by (x+6):
7(x+6)≥8(x+5)
Distribute 7 into x+6 and distribute 8 into x+5:
7x+42≥8x+40
Subtract 7x from both sides:
42≥x+40
Subtract 40 from both sides:
2≥x
Hence:
⇒x≤2
Now we have to assess the actual inequality for values that x cannot be due to the rational terms being undefined.
Looking at the denominator x+5, we see x≠−5.
Looking at the denominator x+6, we see x≠−6.
So the final result is:
{x<−6}∪{−5<x≤2}
=======================EDIT=======================
Should mention what Douglas mentioned in the comments to make this result clearer.
After we have determined that x≠−5 and x≠−6, you should investigate the regions between this "critical" points to determine if that region is included in the final solution. In this case, the example Douglas gives of −5.5 in the comments suffices to show that the region −6<x<−5 is not a part of the solution.
