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How to solve the simplest inequalities – one-step inequalities

one-step inequalities

One-step inequalities

One-step inequalities are pretty much self-explanatory – they are the type of inequalities that are solvable in a single step. Most of time, they’ll come in one of these forms:

x+b<c, x+b \geqslant c, x+b>c,  x+b \leqslant c

Since we already learned what inequalities are, and we know the rules and symbols, we’ll concentrate on learning to solve one-step inequalities through several examples. So, let’s get started!

For example 1:

Find set for:

x + 3 < 5
To solve any inequality, we need to “isolate the variable on one side”. In this case, we can do it in two ways. We can either subtract the number 3 from both sides and then solve the expression on the right side.
Subtracting from both sides:

x + 3 < 5 /+(-3)

x + 3 - 3 < 5 - 3
x < 2
On the number line the solution is:
If we multiplying innequalities with negative number, inequality sign will be changed.

Now that we’ve calculated the result, we can present it in two ways: by writing it down as an interval and/or by marking it on the number line. For practice reasons, we’ll do it both ways.

So, this is how we would write down this result as an interval:

x \varepsilon <- \infty, 2>

And this is how we would mark it on the number line:

 

greater than equal less than

 

Example 2:

Let’s try one with multiplication. How would we solve this problem?

\frac{x}{2} \ge -5

\frac{x}{2} \ge -5 \mid \cdot 2

x \ge -\frac{5}{2}

As we can see, the only thing we needed to do was to multiply the whole inequality by the number 2. The solution of our inequality contains all numbers greater than number -\frac{5}{2}, as well as the number -\frac{5}{2} itself. This is due to the presence of the “greater or equal” sign in the inequality. In the form of an interval, the solution would be written down as:

x \varepsilon  [-\frac{5}{2}, \infty>

And like this on the number line:

Example 3:

Let’s try one that requires division, but we’ll make it a bit more interesting. How would we solve this problem?

- 2x > - 8

- 2x > - 8 \mid : (-2)

x < 4

As we already said, a single division was required to solve this inequality, but this example required us to remember a very important information: when the variable changes signs, the inequality sign changes to its opposite as well! So, instead of a “greater than”, we end up with a “lesser than” at the end of our problem!

But all other things stay the same, so the solution looks like this in interval form:

x \varepsilon <- \infty, 4>

And like this on the number line:

So, this is it for one-step inequalities. If you would like to practice some more, feel free to use the worksheets below.

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