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-8z+1>-9z-8
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Add 9z to both sides. You will get an equivalent inequality
-8z + 9z + 1 > -9z + 9z - 8, or
z + 1 > -8.
Add -1 to both sides. You will get an equivalent inequality
z > -9.
Answer. The solution of the original inequality is z > -9.