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 the time, they’ll come in one of these forms:

$$x+b<c$$

$$x+b>c$$

$$x+b \leqslant c$$

$$x+b \geqslant c$$

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

Example 1:

Find the set of solutions for following inequality:

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

$$ x + 3 < 5 /+(-3)$$

$$ x + 3 – 3 < 5 – 3$$$$ x < 2$$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.On the number line, the solution is:So, this is how we would write down this result as an interval:

$ x \varepsilon \left<- \infty, 2\right>$

 

 

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

Note: If we multiply inequalities by negative number, inequality sign changes.

Example 2:

$\frac{x}{2} \ge -\frac{5}{4}$

Solution:

$\frac{x}{2} \ge -\frac{5}{4} \mid \cdot 2$

$ x \ge -\frac{10}{4}$

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

As we can see, the only thing we needed to do was to multiply the whole inequality by 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 than or equal” sign in the inequality. In the form of an interval, the solution would be written down as:

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

and like this on the number line:

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

Example 3:

$ – 2x > – 8$

Solution:

$ – 2x > – 8 \mid : (-2)$

$ x < 4$

As we already said, a single division was required to solve this inequality, but in this example we had to remember a very important information: when the variable changes its sign, the inequality sign changes to its opposite as well! So, instead of a “greater than” sign, we ended up with a “less than” sign.

Therefore, the solution in interval form is :

$ x \varepsilon \left<- \infty, 4\right>$

On the number line:

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