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:
−12<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.
−74>x
Now that we've solved for the inequalities, let's graph them:
−12<x
−74>x
When graphing, start with the number that x is being compared to. For the first inequality, plot (−12,0). 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 −12, all that you'll be doing now is showing the rest of the inequality. To do this, look at the inequality symbol. −12<x tells us that −12is≤ssthanx,soeveryth∈g→theright(th∈kofaνmberl∈e)ofthe-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.
−74>x
Plot −74, then shade everything to the left of −74. 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]}
