What is the smallest integer that makes $-3x+7-5x<15?$

Hint: We will transpose the terms accordingly to solve the given inequality. We will use the polynomial addition to simplify the terms. Then we will change the sign if necessary and we know that if the sign changes then the inequality changes.

Complete step by step answer:
Let us consider the given inequality $-3x+7-5x<15.$
Let us use the polynomial addition to simplify the left-hand side of the inequality.
Then we will get the inequality as $-8x+7<15.$
Now, we will have to transpose the constant term from the left-hand side of the inequality to the right-hand side of the inequality so that the constant terms lie on the same side and the variable terms on the other side.
Then, we will get the inequality as $-8x<15-7.$
Now, we know that $15-7=8$ and so the right-hand side of the inequality will become $8$ after the operation is done.
So, we will get $-8x<8.$
Now let us transpose $8$ from the left-hand side of the inequality from being the coefficient of the variable to the right-hand side of the inequality.
Then, the inequality will become $-x<\dfrac{8}{8}.$
We know that $\dfrac{8}{8}=1.$ So, we will get the inequality as $-x<1.$
Now, let us multiply the whole equation with a negative one, then, as we know, the inequality will change. So, the inequality will change from ‘less than’ to ‘greater than’ and the left-hand side will become $x$ and the right-hand side will become $-1.$
So, we will get the inequality as $x>-1.$
From this, we will learn that $x$ should always be greater than $-1.$
We know that the smallest integer greater than $-1$ is $0.$
Hence the smallest integer that makes the given inequality true is $0.$

Note: We should not forget to change the inequalities when we multiply the inequality with a negative number. When we multiply an inequality with a negative number, $<$ and $\le $ should be changed to $>$ and $\ge $ respectively and vice versa.