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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} \ge  \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} \ge  \frac{(5x)}{4} /\cdot 4

- 2x \ge  5x  /+(-5x)
- 2x - 5x \ge  0 

- 7x \ge  0 /(-7) 
- 7 < 0
x \le 0

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

x \le 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} \ge  1

(your first instinct here would be to multiply whole inequality with x to get 2 \ge  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} \ge  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.



x < 0: (the inequality sign changes)

2 \le x
x \ge  2
x \in [2, +\infty >


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

2 \ge  x
x \le 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} \le \frac{5}{4} / \cdot (3)

x \le \frac{15}{4}

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


Example 6:

0.3x - \frac{14}{5} \le - 2.8

0.3x - \frac{14}{5} \le \frac{28}{10}

0.3x \le \frac{14}{5} - \frac{14}{5}

0.3x \le 0 / : 0.3

x \le 0

x \in <-\infty, 0 ]


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