# How do you solve and graph 4> -2x+3 or 12< -4x+5?

Jun 5, 2018

$- \frac{1}{2} < x$
$- \frac{7}{4} > x$

#### Explanation:

$4 > - 2 x + 3$
$12 < - 4 x + 5$

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:

$4 > - 2 x + 3$

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:

$1 > - 2 x$

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:

$- \frac{1}{2} < x$

Let's do the other one:

$12 < - 4 x + 5$

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:

$7 < - 4 x$

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.

$- \frac{7}{4} > x$

Now that we've solved for the inequalities, let's graph them:

$- \frac{1}{2} < x$

$- \frac{7}{4} > x$

When graphing, start with the number that $x$ is being compared to. For the first inequality, plot $\left(- \frac{1}{2} , 0\right) .$ 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 $- \frac{1}{2}$, all that you'll be doing now is showing the rest of the inequality. To do this, look at the inequality symbol. $- \frac{1}{2} < x$ tells us that $- \frac{1}{2} i s \le s s t h a n$x$, s o e v e r y t h \in g \to t h e r i g h t \left(t h \in k o f a \nu m b e r l \in e\right) o f t h e$-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.

$- \frac{7}{4} > x$

Plot $- \frac{7}{4}$, then shade everything to the left of $- \frac{7}{4}$. 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]}