First-degree inequalities. Exercises resolved step by step.

Here I will explain step by step how to solve the first degree inequalities, solving exercises step by step. We will see how to solve the first degree inequalities with one unknown and two unknown.

How to solve first-degree inequalities with an unknown

First-degree equations with one question are solved almost the same as first-degree equations. Therefore, it is essential to know how to solve first degree equations.

There are two differences regarding solving first degree equations:

The first difference is that when we multiply or divide a negative number from one member to another, inequality changes direction.

And be very careful, because inequality does not change direction for terms that are adding or subtracting.

The second difference is the form of the solution: whereas in a first degree equation, the solution is a single point, in a first degree equation, the solution is a range of values.

Let’s look at it with a couple of examples of some basic first-degree equations:

inequalities exercises solved step by step

We start by solving this mismatch, passing the terms with x to the first member:

first degree equations solved exercises

Notice that by passing -2x to the first member, the sense of inequality has not changed, as it goes from subtracting to adding.

We group terms in the first member:

exercises of solved inequalities step by step

Now, the 3 that is multiplying the x, passes dividing the 6 into the second term. As 3 is positive, it does not change the direction of inequality either:

first degree equations solved step by step

And finally we solve the division:

solved exercises of first degree inequalities

The solution to the mismatch is the values of x greater than 2, not including it, or in other words, the values of x belonging to the open interval between 2 and infinity:

inequalities exercises solved step by step

The solution represented on the line is as follows:

Let’s see another example:

first degree equation exercises

We pass the terms with x to the first member and the numbers to the second member (the sense of inequality remains):

exercises of first degree inequalities solved step by step

We grouped terms:

first degree inequalities

We have a negative number left multiplying the x by the first member. To clear the x, move on to the second member by dividing and being a negative number, it changes the direction of the inequality. Therefore, in addition to passing the-4 by dividing, we turn inequality around:

linear equations exercises solved step by step

All that remains now is to solve the division:

inequalities solved exercises

The solution to the inequality is all the values of x that are less than or equal to -5, including -5, since in inequality we have the equal sign. Another way to indicate the solution is to say that they are the values of x belonging to the interval opened on the left and closed on the right, from less infinite to -5.

The number of the solution is included when the equal sign appears in the inequality, i.e. whether the inequality is “greater or equal” or “less or equal”.

The solution is represented as follows on the number line:

How to solve first degree inequalities with denominators

Let’s now look at how to solve first-degree inequalities with denominators, while solving this example:

first degree equations with an unknown

To begin with, we reduce it to a common denominator:

solved first degree equations

We removed the denominators:

solved inequalities step by step

We’ll take care of the parenthesis:

solved inequalities step by step

We pass the terms with x to the first member and the numbers to the second member:

first degree equations with an unknown

We grouped terms:

solved first degree inequalities

Now, we have to clear the x, passing the -5 to the second member dividing. As we pass a negative number to the second member, we turn the inequality around:

first degree equation exercises solved

The solution is the range of x-values greater than -411/5, or the same thing:

first degree equations exercises

The values of x belonging to the open interval between -411/5 and infinity.

On the real line it is represented as follows:

How to solve first degree inequalities with two unknowns

Once we are clear on how to solve first degree inequalities with one question, we will see how to find the solution to first degree inequalities with two questions.

In this case, the solution is no longer a range of values of x, but a half-plane, limited by the line that results from passing the equation as an equation (we will see how to do this a little further down).

For example, we are going to solve this inequality with two unknowns:

exercises on first degree inequalities

First we turn inequality into an equation by changing inequality into an equal sign:

exercises of solved inequalities

We have the equation of a line left. The equation of a line has this general shape:

For our equation as well as the general shape of a line, we clear the “y” and it fits:

solved problems of first degree inequalities

At this point, we have to represent that line in the coordinate axes.

To do this, we have to give values to the x and y to obtain their corresponding “y” values, which will correspond to coordinates of points belonging to the line. For example, we are going to give the values 0 and 1 to x.

When x is equal to 0, we replace x with 0 and we obtain the value of “y”:

first degree equations with a variable solved exercises

We obtain a value of y=2, therefore, the point we have obtained, which we will call A is (0,2)

exercises solved from inequalities

We do the same thing when x equals 1:

first degree equation exercises

Gives us a value of y=-1. This time, we have obtained the point (1,-1), which we will call B:

exercises with first degree inequalities

Once we have these two points, we represent them on the Cartesian plane:

first degree equations exercises to solve

And we put those two points together to get the line we wanted to represent:

examples of first degree inequalities

This line divides the Cartesian plane into two half-planes. One is above the line (or in this case on the right) and the other is below it (or in this case on the left).

One of the two half-planes will be the solution to the inequity.

How do we know which semi-flat is the solution?

A point is chosen that is outside the line, if it meets the equality, it means that the half-plane in which the point is located is the solution of the inequality. If it is not met, it means that the solution is the other half flat.

Let’s see how to do this step.

For example, we choose the point (1,1), which does not belong to the straight line and is in the half plane of the right and we substitute its values of x and y in the inequality to check if it is fulfilled or not.

Therefore, in our inequality:

first degree equations examples

We replace the x with 1 and the y with 1:

first degree inequality examples

We operate and see that the inequality is met, since 4 is greater than 2:

resolved exercises of first degree inequalities

Therefore, the half-plane where the point (1.1) is, that is, the one to the right of the line is the solution of the inequity.

But we haven’t finished finding the solution yet.

It is true that 4 is greater than 2, but our inequality is “greater or equal”. What does that mean?

When we have the equal sign in inequality it means that the line itself is also part of the solution. If inequality did not have the equal sign, the line does not belong to the solution.

Finally, all that remains for us to do is to express the solution. The only way to express it is to represent it graphically on the Cartesian plane, since it is not possible to represent it by intervals.

The solution is represented as follows (everything in blue is part of the solution):

first degree equations

Any point that belongs to that half-plane will fulfill the inequality.