Created by Maciej Kowalski, PhD candidate Show
Reviewed by Steven Wooding Last updated: Apr 06, 2022 Welcome to Omni's graphing inequalities on a number line calculator, where we'll take on some linear inequalities and see how to plot them on the number line. And once we see how to deal with one, we'll add some more to the pile and get to graphing systems of inequalities. The resulting number line graph will be a simple tool to find values that satisfy all given conditions: something that our graphing compound inequalities calculator provides as well. Linear inequalitiesInequalities in math are numerical relations that describe where a value lies with respect to some other one. By default, there are four of them: they say whether the first number is:
the second number. For instance: 123 < 216 ,states that Such relations seem straightforward when done on numbers. Well, fair enough: it gets a bit more tricky with negative numbers or decimals, but in the end, it still boils down to a few simple rules. It gets more interesting if we decide to include a bit of algebra and introduce variables. We know that equations are formulas that some unknown variables have to satisfy. Inequalities are similar. However, instead of being specific (e.g., saying that multiplying the number by some value will give a concrete result), they give a rough idea of where the result lies. For instance: x - 10 > 13 ,states that
subtracting Linear inequalities are those with variables only in the first power. They cannot have them in higher exponents, in the denominator of a quotient, under a cube root, inside a logarithmic function, etc. Below, you can find several examples of linear inequalities.
Note how more variables can still give linear inequalities as long as they satisfy the restrictions above. However, we'll focus here only on those with a single one, which we'll denote by But that, folks, is a topic for a separate section. How to graph inequalitiesBefore we see how to graph, say, the inequality of greater than or equal to on a number line, let's spare a few words about the number line graph itself. We can order all real numbers and mark them on an infinite axis called the number line. In essence, the line tells us where one value lies with respect to others: is it larger (to the right) or smaller (to the left) of something else? If we recall the relations mentioned in the first section, it indeed seems like the perfect tool for representing inequalities visually. The first lesson on how to graph inequalities is: you have to solve them first. To be precise, we need to go from expressions like: 3x + 1 ≥ 7 ,to something of the form: x ≥ 2 ,i.e., a single Our topic here is the graphing compound inequalities calculator, so we'll skip instructions on how to solve such things. Let us only briefly mention that we do it the same way we deal with ordinary equations, with the exception that we need to change the inequality sign whenever we multiply or divide by a negative number. Let's assume that we obtained something
like
Note: there exists another popular way for distinguishing strict and non-strict inequalities (point 3 above). Namely, we draw a small circle centered at the point from the inequality and either:
Here, we use the variant in point 3 above since it's what Omni's graphing inequalities on a number line calculator uses. Remember that the number line graph is a representation of an axis that is infinite in both directions. As such, the linear inequality plots also are
infinite (but in one direction). Therefore, they mark sets of all numbers from minus infinity to some value (for Now that we know how to graph inequalities on a number, what do you suspect our next move should be? That's right: graph even more of them. Graphing systems of inequalitiesAs mentioned in the first section, linear inequalities give us a rough idea of what values our variable admits. Sometimes, we might want to limit the possibilities even further, so we introduce yet another inequality and demand that the variable satisfies both of them. And if we're still unsatisfied, we can have yet another. And another. Systems of inequalities (like systems of equations) are lists of inequalities that we want all to be true simultaneously. They are also called compound inequalities. Obviously, introducing a new inequality may ruin the first one or change nothing. For instance, the system: x < 3 , x > 6 ,has no solutions. On the other hand, in the system: x < 3 , x < 6 ,the second inequality is pointless since every number smaller than In essence, graphing systems of inequalities is easy. We simply draw them one by one on the same number line graph. Mind you, we recommend using different colors so that you don't mix them up. Child's play, wouldn't you say? Reading off the result is more tricky. To find what numbers satisfy all the relations simultaneously, we need the values that fall under each drawn line. Usually, we shade the area under the drawn lines (again, with different colors for each) and check where all the colors meet. Also, remember that the greater than or equal to on a number line admits the value it starts with, so make sure to distinguish between strict and non-strict inequalities when you consider the limit points of your solution set. Well, reading about drawings and colors might not be enough to explain the topic properly. Why don't we try out the graphing inequalities on a number line calculator and put our crayons to good use? Example: using the graphing inequalities on a number line calculatorLet's see how to graph inequalities in practice by drawing the number line graph for the following system of inequalities: x < 2 , x ≥ -1 .However, before we grab the crayons, let's see how easy the task is with Omni's graphing inequalities on a number line calculator. Firstly, we tell the tool how many inequalities we have. In our case, there are two, so we select The moment we write the second number, the graphing compound inequalities calculator will show us a graph underneath. Note how the tool also spits out the solution to your system of inequalities in interval notation below the graph. Now, let's describe how to graph the inequalities ourselves. We begin by drawing a number line and marking the two points given by the inequalities: in our case, they are All in all, we obtain the following number line graph: We see that the two colors overlap in between Remember that Omni's graphing inequalities on a number line calculator allows up to three compound inequalities, so we can add one more to the system above. Make sure to play around with the tool and check out other arithmetic calculators we have on offer. FAQHow do I graph solutions to inequalities on a number line?To graph solutions to inequalities on a number line, you need to:
How do I graph compound inequalities on a number line?To graph compound inequalities on a number line, you need to:
How do I graph inequality equations on a number line?To graph inequality equations on a number line, you need to:
How do I read inequalities on a number line?To read inequalities on a number line, you need to:
Maciej Kowalski, PhD candidate Absolute value equationAbsolute value inequalitiesAdding and subtracting polynomials… 33 more How do you graph an equation on a number line?Just follow these steps.. Find the number on the other side of the inequality sign from the variable (like the 4 in x > 4).. Sketch a number line and draw an open circle around that number.. Fill in the circle if and only if the variable can also equal that number.. Shade all numbers the variable can be.. How do you graph a set of solutions?To represent the solution set of a linear equation, y = mx + b, we graph the equation using the following steps: Find two or more points that satisfy the equation. Plot those points on a graph. Connect the points in a straight line.
How do you graph a solution to an inequality on a number line?To plot an inequality, such as x>3, on a number line, first draw a circle over the number (e.g., 3). Then if the sign includes equal to (≥ or ≤), fill in the circle. If the sign does not include equal to (> or <), leave the circle unfilled in.
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