There are many forms of **multi-step inequalities**, but they all can be reduced to a few simple forms, so we’ll start with examples and learn along the way.

We must remember how we solve equations:

It is valid:$\forall a, b, c$

- if $a<b$, then $\forall c\in \mathbb{R}$:

$a+c<b+c$ (1)

- if $a<b$, then $\forall c\in \mathbb{R}$:

$a \cdot c<b \cdot c$ (2)

- if $a<b$ and $b<c$

$a<c$ (3)

**Example 1:** Solve inequality and present it graphically:

Now we have our variable $x$ on both sides. First we add what we can. That means, free numbers with free numbers, and numbers with $x$.

*Free numbers (variables) cannot be added with numbers with*$x$ simply because you don’t know what your variable $x$ is (can’t mix apples and oranges).

So the $x$ on the right side comes to the left from the inequality sign, and $-1$ goes from left to the right (__both of them have to change their sign!__). And then we have:

$ x > 4$

$ x \in <4, +\infty>$

**Example 2**: Solve multi-step inequality and present it graphically:

First we multiply by $5$ and $2$ “to get rid of the braces”. By doing that we get:

From this point, you just repeat the steps from

*Example 1.*

$ 3x > 9 /: 3$

$ x > 3$

Of course things can be made a bit more complicated with fractions, so let’s do that.

**Example 3**: Solve inequality and present it graphically:

First step is to look at it and see if there is $x$ in the denominators. If it is not, it is safe to multiply (just be careful about signs). Common denominator of these two fractions is number $4$, so we’ll multiply whole inequality by $4$.

$ – 2x ≥ 5x /+(-5x)$

$ – 2x – 5x ≥ 0$

$- 7x ≥ 0 /(-7)$

$ x ≤ 0$

We got our variable $x$, now just remains to present it graphically.

$ x ≤ 0$

There is one more complication that may occur. That is finding x in your denominator. As you know, you cannot divide with zero. Since there is a possibility that in your calculation you include that zero, that is a big mistake. So the first thing you do, when you see that kind of a task, is that you take care of the denominators, i.e. to exclude things that cannot be.

Let’s see it on a **Example 4:**

*(your first instinct here would be to multiply whole inequality with*$x$

*to get*$ 2 ≥ x$

*, but you have to be careful, first, you have to exclude cases where you divide by zero. Here you have to set the condition that*$x$

*must not be*$0$

*, because if*$x = 0$

*, you have*$2:0$

*, and that can’t happen)*So your condition is that $ x \not= 0$.

And now you can multiply!

How do we treat our inequality sign? Is variable $x$ greater or lesser than zero? He can be both as far as we know. So we’ll divide this into two cases.

**1.**

$ x < 0$**: **(the inequality sign changes)

**2.**

$x > 0$**: **(the inequality sign remains the same)

This is how we write it then: $ x \in <-\infty, 2 ] \backslash 0$.

On the number line it would look like this:

From the picture we can conclude that our solution is whole set of real numbers, just without zero.

We write that like this: $ x \in \mathbb{R} \backslash 0$.

**Example 5:**

$ x ≤ \frac{15}{4}$

$\frac{15}{4}= 3 \frac{3}{4}$

**Example 6:**

$ 0.3x – \frac{14}{5} ≤ \frac{28}{10}$

$ 0.3x ≤ \frac{14}{5} – \frac{14}{5}$

$0.3x ≤ 0 / : 0.3$

$ x ≤ 0$

$ x \in <-\infty, 0 ]$

## Multi-step inequalities worksheets

**Solve integers** (435.4 KiB, 1,217 hits)

**Solve decimals** (470.3 KiB, 943 hits)

**Solve fractions** (591.0 KiB, 1,045 hits)