# Multi-Step Inequalities

## Key Questions

Inequalities are very tricky.

#### Explanation:

When solving a multi step equation, you use PEMDAS (parentheses, exponents, multiplication, division, add, subtract), and you also use PEMDAS when solving a multi step inequality. However, inequalities are tricky in the fact that if you multiply or divide by a negative number, you must flip the sign. And while normally there are 1 or 2 solutions to a multi step equation, in the form of x= , you'll have the same thing, but with an inequality sign (or signs).

• I would start with arranging terms so that all variables are on one side.

I hope that this was helpful.

There are generally 3 methods to solve inequalities

#### Explanation:

We can usually solve inequalities by 3 ways:

1. By algebraic method
Example 1: Solve: 2x - 7 < x - 5
2x - x < 7 - 5
x < 2
2. By the number-line method.
Example 2. Solve $f \left(x\right) = {x}^{2} + 2 x - 3 < 0$
First, solve f(x) = 0. There are 2 real roots x1 = 1, and x2 = - 3.
Replace x = 0 into f(x). We find f(0) = - 3 < 0. Therefor, the origin O is located inside the solution set.
Answer by interval: (- 3, 1)

--------------------- - 3 ++++++++ 0 ++++ 1 ----------------

3.By graphing method.
Example 2. Solve: $f \left(x\right) = {x}^{2} + 2 x - 3 < 0$.
The graph of f(x) is an upward parabola (a > 0), that intersects the x-axis at x1 = 1 and x2 = - 3. Inside the interval (-3, 1), the parabola stays below the x-axis --> f(x) < 0.
Therefor, the solution set is the open interval (-3, 1)
graph{x^2 + 2x - 3 [-10, 10, -5, 5]}

• Suppose we're solving in $\setminus m a t h \boldsymbol{R}$

$0 x = 1 \setminus R i g h t a r r o w S = \setminus \emptyset$

$0 x = 0 \setminus R i g h t a r r o w S = \setminus m a t h \boldsymbol{R}$

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