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:

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.