Solving Combined Inequalities

Example:

Solve and plot the following combined inequality \quad -6<2x+3\leq 10

To solve a combined inequality, we need to split it up. We’ll start with the left side.

-6 < 2x+3

-6 + (-3) < 2x + 3 + (-3)

(\frac{-9}{2}) = (\frac{2}{2})x

x > -4\frac{1}{2}

Now we can solve the right side of our inequality.

2x+3\leq 10

3x+3+(-3)\leq 10 + (-3)

(\frac{2}{2})\leq \frac{7}{2}

x\leq 7/2

x \leq 3 (\frac{1}{2})

 

So our final inequality is: \quad -4(\frac{1}{2}) < x \leq 3(\frac{1}{2})