One-Variable Linear Inequalities & Formatting Their Solutions

Then, after combining the like terms, you divided through by whatever was multiplied onto the variable, to get (variable) equals (a number).

Solving one-variable linear inequalities is almost exactly like solving the one-variable linear equations. But, where the solutions to linear equations are single values, the solutions to linear inequalities are infinite intervals.

Rather than having, say, a solution of "x = 2" for a linear equation, you will have a solution of, say, "x ≤ 2" for a linear inequality. And while this "less than or equal to" solution has a boundary value (that is, it has a specific edge-point of the solution) of x = 2, the solution to the in-equation (that is, to the inequality) itself contains infinitely-many points, being every single real number to the left of 2.

• Solve x + 3 < 0

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What are the 4 formats for solutions to inequalities?

Solutions to one-variable linear inequalities can be formatted in any of four ways. Using the inequality x < −3 for our examples, these formats are:

• Inequality notation: x < −3
• Set notation: {x | x < −3}
• Interval notation: (−∞, −3)
• Graphing: shading (thickening) a number line

In the exercise I did above, my solution was formatted in inequality notation, so-called because the solution was written as an inequality. This is probably the simplest of the solution notations, and the most natural to use, but there are three other notations with which you might need to be familiar.

A solution in "set notation" writes the solution as a set of points. The above solution would be written in set notation as:

...which is pronounced as "the set of all x-values, such that x is a real number, and x is less than minus three".

A simpler form of this notation would be something like:

...which is pronounced as "all x such that x is less than minus three". You might even see set-notation solutions stated as the inequality solution, but that inequality will be put inside curly braces to say that the solution is a set of points: {x < −3}.

"Interval notation" writes the solution as an interval (that is, as a section or length along the number line). The above solution, x < −3, would be written as:

...which is pronounced as "the interval from negative infinity to minus three", or just "minus infinity to minus three".

Interval notation can be easier to write than to pronounce, because of the ambiguity regarding whether or not the endpoints are included in the interval. (Note: Infinity is *never* included as an endpoint; infinity is not a "point" which can be "reached".) To denote, for instance, the inequality solution "x ≤ −3", the interval would be written as:

...which is pronounced as "minus infinity through (not just "to") minus three" or "minus infinity to minus three, inclusive", meaning that −3 is to be included within the solution. The right-parenthesis in the interval form of the solution x < −3 indicated that the −3 was not included; the right-bracket in the interval form of x ≤ −3 indicates that it is included.

The last form of solution notation is actually more of an illustration. You may be directed to "graph" the solution. This means that you would draw the number line, and then highlight the portion that is included in the solution to the inequality. First, you would mark off the edge of the solution interval, in our example being the point −3. Since −3 is not included in the solution (because this is a "less than" inequality, not a "less than or equal to" inequality), you would mark this point with an open dot or with an open parenthesis pointing in the direction of the rest of the solution interval:

...or:

Then you would shade in the appropriate side:

...or:

Why shade to the left? Because they want all the values that are less than −3, and those values are to the left of the boundary point. If they had wanted the "greater than" points, then I would have shaded to the right.

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In all, we have seen four ways, with a couple variants, to denote the solution to the above inequality:

• inequality notation:
  • written: x < −3
  • spoken: x is less than minus three
• set notation:
  • written:
    1. {x | x is a real number, x < −3}, or
       
    2. {x | x < −3}, or simply {x < −3}
  • spoken:
    1. the set of all x, such that x is a real number and x is less than minus three
    2. all x such that x is less than minus three
• interval notation:
  • written: (−∞, −3)
  • spoken: the interval from minus infinity to minus three
• graphing notation:
  • written (using an open dot, or a parenthesis):
    • 
    • 
  • spoken: minus infinity to minus three

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The notation is slightly different when the inequality is of the "or equal to" variety, so I'll list them all for the following example:

• Solve x − 4 ≥ 0

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Just as before, this solution (namely, x ≥ 4) can be presented in any of the four following ways:

• inequality notation:
  • written: x ≥ 4
  • spoken: x is greater than or equal to four
• set notation:
  • written:
    1. {x | x is a real number, x ≥ 4}, or
       
    2. {x | x ≥ 4}, or simply {x ≥ 4}
  • spoken:
    1. the set of all x, such that x is a real number and x is greater than or equal to four
    2. all x such that x is greater than or equal to four
• interval notation:
  • written: [4, ∞)
  • spoken: the interval from four to positive infinity, inclusive
• graphing notation:
  • written (using a closed dot, or a bracket):
    • 
    • 
  • spoken: four to plus infinity, inclusive

When you're doing a graph of a solution, the square bracket notation goes with the parenthesis notation, and the closed (that is, the filled in) dot notation goes with the open dot notation.

If (further on) you have an inequality where both endpoints are actual numbers (that is, where neither endpoint is −∞ or +∞), and one endpoint is included while the other is not, you'll want to use matching symbols: one parenthesis and one bracket, or one open dot and one closed dot. Don't mix the symbols. I don't know that it would be wrong, per se, but it will surely annoy the grader.

While your present textbook may require that you know only one or two of the above formats for your answers, this topic of inequalities tends to arise in other contexts in other books for other courses. Since you may need later to be able to understand the other formats, make sure now that you know them all. However, for the rest of this lesson, I'll use only the inequality notation.

I like it best.

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By the way, your textbook may also have you "test to see if a value is a solution". This means that they'll give you an inequality, along with a number, and they're expecting you to plug the number into the inequality, and see if the number works. For instance:

• Is 2 a solution to x ≤ 5?

On the other hand:

• Is 2 a solution to x > 5?

Of course, there are infinitely-many other points which are (or are not) solutions to any given inequality. So you won't be seeing much of this sort of exercise.