When solving inequalities using multiplication, explain the difference between multiplying each side of the equation by a positive number and multiplying each side of the equation by a negative number.

Question

anonymous · Answer

when you multiply by negative you have to change the direction 
but when you multiply by positive you keep the direction

ciarán95 · Answer

The best way to do this is to use a simple inequality example: $$2 < 3$$ We know this is true, and if we were to multiply each side of the equation by a positive number then we would get: $$2(2) < 2(3)$$, or $$4<6$$, which still holds,as it will for multiplying both sides by any positive number. If we were to multiply by -2, however, we get: $$-2(2) < -2(3)$$, or $$-4 < -6$$, which isn't true. So if we multiply both sides by a negative value, we must flip the direction of the inequality (this happens whether it was 'less than' or 'less than or equal to'). So, we get $$-4>-6$$, which is now true. If we had an inequality $$-x > 4 - 5$$, and we wanted to get rid of the minus sign in front of the x to solve for x on its own, then we could multiply each side by -1. However, we would have to change '>' to a '<'. Thus, $$x < -1(4 - 5)$$, which solves as $$x<1$$