#4 > -2x + 3#
#12 < -4x + 5#
First, let's recall that solving inequalities works similar to equations that you're already familiar with. In this case, these two inequalities are two-step equations, which means you have two steps to solve them. Let's do each one, step by step:
#4 > -2x + 3#
Subtract #3# from both sides to cancel out positive #3#. This is the first step, and will help you get closer to isolating for the value of #x#. Your equation should now look like this:
#1 > -2x#
Now, divide by #-2# in order to isolate for the value of #x#. Note that when you divide by a negative number, the inequality sign "flips", or changes to its opposite. We'll go through how these a graphed in a moment. Your equation should now look like this:
#-1/2 < x#
Let's do the other one:
#12 < -4x + 5#
Subtract #5# from both sides to cancel out positive #5#. This is similar to how you solved the first inequality. Your equation should now look like this:
#7 < -4x#
Divide by #-4# to isolate for the value of #x#. Recall that, again, the sign will "flip", or change to its opposite when divided by a negative number.
#-7/4 > x#
Now that we've solved for the inequalities, let's graph them:
#-1/2 < x#
#-7/4 > x#
When graphing, start with the number that #x# is being compared to. For the first inequality, plot #(-1/2 , 0).# Here's how to remember it: inequalities dealing with #x# are on the #x#-axis, and inequalities dealing with #y# are on the #y#-axis.
Now that you've plotted #-1/2#, all that you'll be doing now is showing the rest of the inequality. To do this, look at the inequality symbol. #-1/2 < x# tells us that #-1/2 is less than #x#, so everything to the right (think of a number line) of the #-1/2# will be shaded. The physical line itself will be dotted because the inequality doesn't include a "greater / less than or equal to" sign. If this were the case, the line would be solid.
#-7/4 > x#
Plot #-7/4#, then shade everything to the left of #-7/4#. The line will be dotted, just like the previous equation.
So:
graph{-1/2 < x [-10, 10, -5, 5]}
graph{-7/4 > x [-10, 10, -5, 5]}
