Omni's absolute value inequalities calculator will help you whenever you stand in front of one of those scary absolute value inequalities problems. In this article, we will mainly talk about:

  • Solving absolute value inequalities algebraically, and
  • Graphing absolute value inequalities.

We will also show a few examples of absolute value inequalities. We bet you will quickly become an expert and start teaching others how to solve absolute value inequalities and absolute value equations!

What are absolute value inequalities?

If you wish to solve absolute value inequalities, you must thoroughly understand the concept of the absolute value. Let's recall that the absolute value of x (which we denote by |x|) is the distance between this number and zero. In other words, if x is a real number, then:

  • |x| = x if x is non-negative, and
  • |x| = -x if x is negative.

For example:

  • |-1| = 1;
  • |0| = 0, and
  • |11| = 11.

As you can see, it's really simple to notice and remember the rule: the absolute value of a non-negative number is just this number, while if the number is negative, we have to remember to erase the minus sign.

Now, absolute value inequality is any inequality that contains the absolute value of some expression. For instance, the inequality |x2 + 3x -18| < 3 involves a quadratic expression. Most often, however, we have to deal with absolute value inequalities containing linear expression, namely bx+c. In the most general form, they can be written as:

a * |bx + c| + d > e,

where the > can be, of course, replaced by <, , or . The most efficient way of dealing with such inequalities is… to use Omni's absolute value inequalities calculator 🙂. In what follows, we explain how it works before discussing how to solve absolute value inequalities by hand.

How to use this absolute value inequalities calculator?

Our absolute value inequalities calculator can solve absolute value inequalities involving the expression a * |bx + c| + d. To do so, follow these steps:

  1. Choose the sign of the inequality: >, ≥, ≤, <, =.
  2. Enter the coefficients a, b, c, d, e of your inequality.
  3. The calculator will display your inequality below the fields - make sure everything is correct.
  4. The solution to your absolute value inequality will appear as well. By default, it will have the precision of two decimal places - click the Advanced mode to adjust it.
  5. Our absolute value inequalities calculator has two extra options:
    • If you want to see intermediate computations and extra explanations, change the Show steps? option to Yes.
    • If you want to see the solution in a graphical form, turn the Show graph? option to Yes.

If you want to better understand how graphing inequalities on a number line works, visit our dedicated calculator.

How to solve absolute value inequalities by hand?

Here's how to do absolute value inequalities in the form a × |bx + c| + d > e or similar:

  1. Simplify your inequality: subtract d from both sides and then divide by a. Remember to flip the sign if a < 0!
  2. If the right-hand side is negative, then your inequality has no solutions. Otherwise, it has at least one solution.
  3. Omit the absolute value, remembering that:
    • If the sign is > (or ), then you get the alternative: bx + c > (e - d)/a or -(bx + c) > (e - d)/a. Solve these inequalities for x. As you can see, fractions may appear.
    • If the sign is < (or ), then you get the conjunction: bx + c < (e - d)/a and -(bx + c) < (e - d)/a. Equivalently: -(e - d)/a < bx + c < (e - d)/a. Again, solve for x.
  4. Finally, you may wish to rewrite the solution in interval notation.

🙋 If you want to learn how to convert between the inequality and interval notation, go to the inequality to interval notation calculator.

As for now, we've discussed how to solve absolute value inequalities algebraically. There's another way, though, which some may find more intuitive. Namely, we will now show you how to graph absolute value inequalities. Then we will go through several absolute value inequalities problems together.

How to graph absolute value inequalities?

To be effective at graphing absolute value inequalities of the form a × |bx + c| + d > e, remember the following rules:

  1. To plot |bx + c|, draw a straight line through the points (0,c) and (-c/b,0) and then reflect its negative part (where bx+c < 0) through the horizontal axis.
  2. To produce a × |bx + c| + d, adjust the slope (multiply by a and, if a<0, reflect through the horizontal axis) as well as move the plot up (if d>0) or down (if d<0). Done!
  3. Draw the horizontal line y = e.
  4. Since our sign is >, the arguments of x where the first plot is strictly above the line y = e are the solutions.
  5. If the sign was <, we'd look for the arguments that are strictly below the line y = e. If the sign was or , then the points of intersection are also solutions.

And that's it when it comes to graphing absolute value inequalities! Well, the theory is nice, you may (and should) say, but it's practice that makes perfect, right? Right! It's high time we worked through a few examples of absolute value inequalities.

Examples of absolute value inequalities

Here you can see real examples of how to do absolute value inequalities. Let's start with simpler ones and then move on to some more challenging computations.

Example 1.

Solve -2 * |x + 3| ≥ 0.

  1. Our inequality is equivalent to |x + 3| ≤ 0. Note, that we invert the inequality sign while dividing by the negative number -2!
  2. Since the absolute value cannot be negative, our inequality is further equivalent to |x + 3| = 0.
  3. Recall that the absolute value of a number is equal to zero if, and only if, this number is zero. So we have x + 3 = 0.
  4. We easily obtain x = -3. Our inequality has only one solution.

Example 2.

Solve |x - 8| + 7 ≥ 6.

  1. Clearly, our inequality is equivalent to |x - 8| ≥ -1.
  2. When is the absolute value greater than or equal to -1? Well, always, because it's always non-negative!
  3. Hence, our inequality holds no matter what value we substitute in for x. In other words, it holds for all real numbers.
  4. In interval notation, our inequality holds for x ∈ (-∞, ∞).

Example 3.

Solve |2x - 3| - 1 < 6.

  1. Our inequality is equivalent to |2x - 3| < 7.
  2. Hence, we have 2x - 3 < 7 and 2x - 3 > -7.
  3. Simplifying, we get x < 5 and x > -2. We can rewrite this as -2 < x < 5.
  4. In interval notation we get x ∈ (-2, 5).

Example 4.

Solve |x + 5| - 1 > 11.

  1. We have |x + 5| > 12. So, the distance between 0 and x+5 must exceed 12.
  2. That is, x + 5 > 12 or x + 5 < -12.
  3. Simplifying, we obtain x > 7 or x < -17.
  4. In interval notation we have x ∈ (-∞, -17) ∪ (7, ∞).

FAQ

How many solutions can an absolute value inequality have?

An absolute value inequality in the form a × |bx + c| + d ≥ e (or ) can have infinitely many solutions, one solution or zero solutions. If the sign is that of a strict inequality, i.e., > or <, then the inequality can have infinitely many solutions or no solutions at all.

When does an absolute value inequality have no solutions?

An absolute value inequality involving a × |bx + c| + d has zero solutions if, after simplification, we obtain an inequality claiming that the absolute value of some expression is negative. As we all know, the absolute value cannot under any circumstances be negative, and so the original inequality cannot have solutions.

Anna Szczepanek, PhD
a ∙ |b∙x + c| + d > e
Sign
>
a
b
c
d
e
Plot the problem?
No
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