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How to solve the simplest 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:

inequalities comparison

 

Those forms are not the only ones, though. We will also encounter inequalities that require a single multiplication or division to get to the result. 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 the x in:

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 or transfer the 3 from the left to the right side of the inequality, and then solve the expression on the right side. Both ways count as a single step. We’ll do it both ways to get more practice.

Subtracting from both sides:

x + 3 < 5

x + 3 - 3 < 5 - 3
x < 2
 

Transfer of the 3:

x + 3 < 5

x < 5 - 3

x < 2

The important thing to remember in this one is that, when you transfer a number or a variable from one side of the inequality to the other, the sign of the number or the variable we transferred changes into its opposite. And as we can see, all numbers that are smaller than 2 are the solution of this inequality.

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:

open interval example

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 * 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:

greater than graph includedExample 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 piece of 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:

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

One-step inequalities worksheets

  Graphing one-step inequalities (1.5 MiB, 176 hits)

  Solve Integers by using addition/subtraction (373.6 KiB, 588 hits)

  Solve Integers by using multiplication/division (384.7 KiB, 433 hits)

  Solve integers (382.8 KiB, 402 hits)

  Solve decimals by using addition/subtraction (385.2 KiB, 398 hits)

  Solve decimals by using multiplication/division (394.5 KiB, 387 hits)

  Solve decimals (397.8 KiB, 409 hits)

  Solve fractions by using addition/subtraction (454.3 KiB, 345 hits)

  Solve fractions by using multiplication/subtraction (428.5 KiB, 220 hits)

  Solve fractions (460.0 KiB, 371 hits)

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