Can An Inequality Have Only One Solution?Īn inequality can have only one solution (that is, a single real value of x as the only solution). The graph of y = x 2 + 8x + 16 (blue curve) is never below the horizontal line y = 0 (red line). Since the graph of y = (x + 4) 2 is always above the graph of y = 0, there is no solution to the inequality (x + 4) 2 < 0. This horizontal line is always below the parabola y = (x – 4) 2 (never above it). The right side is a horizontal line at y = 0. Note that this parabola never goes below the x-axis (since the vertex is its lowest point). The left side is a parabola y = (x + 4) 2, with its vertex at the point (-4, 0). So, we can never have (x + 4) 2 < 0, since that would imply (x – 4) 2 is negative. Example 2: Quadratic Inequality With No Real Solution The graph of y = x 2 (blue curve) is never below the horizontal line y = -3 (red line). Since the graph of y = x 2 is always above the graph of y = -3, there is no solution to the inequality x 2 <= -3. This horizontal line is always below the parabola y = x 2 (never above it). The right side is a horizontal line at y = -3. The left side is a parabola y = x 2, with its vertex at the origin (0, 0). So, we can never have x 2 <= -3, since that would imply x 2 is negative.Īnother way to see that there is no solution is to graph both sides of the inequality to see that they never intersect. The “OR” word can give us a larger solution set, while “AND” can give us a smaller solution set.įor instance, x > 2 OR x 2 AND x 0 AND x 0) and negative (x 4 AND x 4) and negative (x 0 Remember that a compound inequality contains linking words such as “OR” & “AND”. So, regardless of which value of x we choose, we can never find one such that |x| > |x| + 1. This last inequality is never true, since 0 is less than 1. It has no solution, which we can see by subtracting |x| from both sides: The absolute value of x, |x|, can never be negative, so it can never satisfy this inequality. That is, when we take the absolute value of any number x, we get zero (for x = 0) or a positive number (for x not equal to zero). Example 1: Absolute Value Inequality With No SolutionĬonsider the following absolute value inequality: With this in mind, let’s take a look at some examples of absolute value inequalities with no solution. You can learn more about absolute value in my article here. This means that |f(x)| >= 0 for any function f(x). You could absolute value inequalities like the following:įor an absolute value inequality, the key thing to remember is that the absolute value symbol gives us a nonnegative number (the result is always positive or zero).

Remember that an absolute value inequality contains the absolute value symbols. Absolute Value Inequalities With No Solution We’ll begin with absolute value inequalities. Let’s take a closer look at each of these cases and some examples. Quadratic Inequalities (with an “x 2” term).Can An Inequality Have No Solution?Īn inequality can have no solution, and there are several cases where this can happen, including: We’ll also talk about when an inequality has one solution or the set of real numbers as its solution set. In this article, we’ll talk about when an inequality can have no solution and how it happens, along with some examples. Of course, we can always find complex numbers to solve quadratic inequalities, but we will only consider real numbers for this discussion. There are also cases where they can have only one solution (a single real number) or the set of all real numbers as solutions. Absolute value inequalities, compound inequalities, and quadratic inequalities can all have no solution in some cases. So, can an inequality have no solution? An inequality can have no solution in several cases. However, you might want to know whether an inequality has solutions and when it may fail to have them. Inequalities are used to compare values and to express ranges or estimates in the case of uncertainty.

No solution graph