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, 1,286 hits)
Solve decimals (470.3 KiB, 1,000 hits)
Solve fractions (591.0 KiB, 1,091 hits)
