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Multi-step inequalities on a number line

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:

$ 2x – 1 > x + 3$

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:

$ 2x – x > 3 + 1$

$ x > 4$
From here, all is familiar: we use the number line, check out our solutions and write them down in the form of a interval.

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

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

$ 5(x – 1) > 2(x + 2)$

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

$ 5x – 5 > 2x + 4$  $/+(5-2x)$

From this point, you just repeat the steps from Example 1.

$ 5x – 2x > 4 + 5$

$ 3x > 9 /: 3$
($ 3 > 0$ inequality remains the same)

$ x > 3$

$ x \in <3, +\infty>$

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:

$\frac{(-x)}{2} ≥ \frac{(5x)}{4}$

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 $4$.

$\frac{(-x)}{2} ≥ \frac{(5x)}{4} /\cdot 4$

$ – 2x ≥ 5x  /+(-5x)$
$ – 2x – 5x ≥ 0$ 

$- 7x ≥ 0 /(-7)$ 
$ – 7 < 0$
$ 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:

$\frac{2}{x} ≥ 1$

(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!

$\frac{2}{x} ≥ 1 / \cdot x$
Now we see another thing that may be a problem.
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$
$x ≥ 2$
$x \in [2, +\infty >$

2.

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

$ 2 ≥ x$
$x ≤ 2$
$ x \in <-\infty, 2 ]$
(Note that 0 is a part of this interval and must be excluded)

 

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:

$ \frac{x}{3} ≤ \frac{5}{4} / \cdot (3)$

$ x ≤ \frac{15}{4}$

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

 

Example 6:

$ 0.3x – \frac{14}{5} ≤ – 2.8$

$ 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, 802 hits)

  Solve decimals (470.3 KiB, 604 hits)

  Solve fractions (591.0 KiB, 676 hits)