In this explainer, we will learn how to solve one-variable quadratic inequalities algebraically and graphically.

Recall that, in an equation, we have two expressions that are equal to each other, and we write the equals sign, , between them. When we have two expressions that are not equal to one another, we can relate the expressions by the use of an inequality sign.

We can have inequalities such as

In each of these inequalities, has a range of possible solutions. When we have an inequality such as , we can say this in words as “ is greater than or equal to four.” This means that a value of that is four or more will satisfy this inequality. The four inequality symbols we use are

We can solve inequalities in a similar process to solving equations, by ensuring that we perform the same mathematical operation to both sides of the inequality. However, as inequalities have a direction, we must be careful to consider which side of the inequality an expression is on. When we multiply or divide by a negative number, we must switch the inequality. For example, if we have then when dividing by we must switch the inequality to give

Let us now look at how to solve an inequality and represent the answer as an interval. Before doing so, we need to recap some notation. If we consider the interval of numbers from 0 to 10, which includes 0 but not 10, we could represent this using inequalities as

The strict inequality on the right tells us that the 10 is not included in the inequality, and the nonstrict inequality on the left tells us that the 0 is included. Another way of writing this interval would be .

Here, the square bracket tells us that the 0 is included, and the curved parenthesis tells us that the 10 is not included. It is also worth recapping here that the symbol for infinity is . This is often used to represent intervals that are greater than or less than a single number. For example, in interval notation would be .

Note that we never put a square bracket next to an infinity symbol as infinity is not a number. We will now look at an example of solving a linear inequality.

### Example 1: Finding the Solution Set of a Linear Inequality

Find the solution set of the inequality . Write your answer as an interval.

First, we subtract 3 from both sides of the inequality , which gives us

We can now divide each side by , remembering that when we divide an inequality by a negative number, we need to switch the inequality sign. This gives us

So is all the numbers greater than or equal to .

To express the solution as an interval, we must represent all the numbers from up to infinity, including . So, we start our interval with a square bracket and finish with the infinity symbol with a parenthesis: .

In the same way that we have distinct equations such as linear and quadratic equations, we can have quadratic inequalities in the following forms.

A quadratic inequality can be in one of the following forms: where , , and are constants and .

When we solve a quadratic inequality, we need to find the range of solutions, or intervals, for which an inequality is true. We can solve quadratic inequalities using the process steps below.

### How to solve a Quadratic Inequality Algebraically

1. Rearrange the inequality so that we have all the terms of the expression, defined as , on one side, with an inequality relating this to zero. For example, or .
2. Solve by factoring, or otherwise, to find the solutions to the equation.
3. Select test points for each interval so that there are values less than, between, and greater than the solutions of the equation. We can also use a sign chart to identify the intervals that will be positive or negative.
4. Identify the intervals that satisfy the inequality.

Let us now look at an example of solving a quadratic inequality using test points to identify the interval solution.

In the next example, we will look at how we can use a sign chart to identify positive and negative values for the intervals of an inequality.

### Example 2: Solving a Quadratic Inequality Using a Sign Chart

Describe all solutions to the inequality .

To begin solving the inequality , we will first transform and rearrange this to get a positive coefficient of . We can multiply all terms of the coefficient by , recalling that when we multiply an inequality by a negative number, we must switch the inequality. This gives us

We now need to solve , where . We can factor our equation to give

Therefore,

So,

To solve the inequality , we need to identify the regions where this is true. Whether or not depends on the signs of the factors and .

We can create a grid to identify whether each factor will be positive or negative in the intervals less than, greater than, and between our solutions of and . In the grid, we can place the intervals between our solutions horizontally and our factors of vertically, with the product of the factors below. We can then calculate whether the product of the factors will be positive or negative.

In the grid above, we can see in the first results column that when , our values and will both be negative, so the product of those two negative values, , will be positive.

Checking the signs of in the grid, we can see that it will be positive, that is, , in the intervals and . We can express our answer as .

### Example 3: Solving a Quadratic Inequality

Determine the solution set of the inequality .

To begin solving this inequality, we will first expand the brackets. Note that we should not take the square root of each side here, as this could lead to an invalid answer.

Expanding the brackets gives us

We now need to collect all the terms on the same side of the inequality. In order to keep a positive coefficient of , we can subtract all the terms on the left-hand side from each side of the inequality, which gives us

Simplifying, we have

We can also write this as noting that the inequality has been switched.

Since 24 is a common factor, we can divide all the terms by 24, which gives us

Setting our and factoring give us

Therefore,

To solve the inequality , we need to identify the regions where this is true. Whether or not depends on the signs of the factors and .

We can create a grid to identify whether each factor will be positive or negative in the intervals less than, greater than, and between our solutions and . Since we have a nonstrict inequality, we can also record the values when and . We can then calculate whether the product of the factors will be positive or negative.

0
0
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From the grid, we can see that the intervals where are when and when . We can write this in interval notation as .

As an alternative method, we could have sketched a graph of . Given that the coefficient of is positive, we know that the parabola curve will open upward. As the roots of the equation are and , this means that the curve will pass through the coordinates and .

To solve the inequality , we consider the points on the graph of , where . This will be above the -axis, at the values where and where . We can express this answer in interval notation as .

In the following example, we will look at an alternative method of solving a quadratic inequality by drawing a graph.

### Example 4: Solving a Quadratic Inequality in a Real-World Context

A cell phone company has the following cost and revenue functions: and where is the number of cell phones.

State the range for the number of cell phones they can produce while making a profit. Round your answers to the nearest integer that guarantees a profit.

In beginning of this question, we need to apply some real-world mathematics. We can calculate profit using the following calculation:

We can define the profit as , and substituting the given functions of cost, , and revenue, , we have

Simplifying the terms, we have

We are asked to find the range for which a profit can be made. This would be a profit greater than 0, which we can write as the inequality

To solve the inequality, we will sketch a graph of . To do this, we first need to solve the equation , to find the points where the equation crosses the -axis. To solve, we will use the quadratic formula.

Recall that a quadratic equation in the form , where , , and are constants, can be solved using the quadratic formula

Therefore, to solve the equation , we can substitute the values , , and into the quadratic formula and simplify to get

Therefore,

Using a calculator, we can evaluate to one decimal place, which gives us

Next, we can sketch a graph, plotting the coordinates of the roots at and . Note that as this is just a sketch to help illustrate the inequality, it does not need to be precise. Since the equation has a coefficient of () that is less than zero, the equation will have a curve that opens downward. Hence, the graph will look as below.

Next we will need to establish the points where . The curve has values greater than zero between the -values of 27.8 and 70.4. This means that the range of cell phones that can be produced while making a profit is between 27.8 and 70.4 cells. However, as we need to give our answers to the nearest integer, our final answer is 28–70 cell phones.

### Example 5: Solving a Quadratic Inequality Using a Graph

Solve the inequality .

To begin solving this inequality, we will perform the same operation to both of its sides. We can subtract from both sides, which gives us

We can then add 27 to both sides of the inequality, giving us

To solve the inequality, we will sketch a graph of . To do this, we first need to find the points where the equation crosses the -axis, often called the roots of the equation.

Setting our , we can factor this, giving us

Therefore,

So,

We now need to establish the shape of the curve . As the coefficient of , 2, is positive, this means that the parabola curve will open upward.

So, as the roots of the equation are and , we can plot the coordinates and and sketch a parabola curve as shown below.

Next, we need to identify the areas for which the inequality holds true. We can see from the sketch that is at values less than zero between the values and . Therefore, this is the interval for the inequality. In interval notation, we can write this as .

Note that as the values can also be exactly equal to both 3 and 4.5, we use square brackets beside each value.

### Example 6: Solving a Quadratic Inequality Using a Graph

Solve the inequality .

To begin solving this inequality, we will first multiply the parentheses, which gives us

We now need to collect all the terms to the same side of the inequality. Adding to both sides, we have

We can then subtract 35 from both sides, which gives

To solve the inequality, we will sketch a graph of . To do this, we first need to find the points where the equation crosses the -axis. These roots can be found by setting and solving, giving us

Factoring, we have

Therefore,

As the coefficient of in the equation is 1, this value is greater than zero; so the parabola curve will open upward. As the roots of the equation are and , this means that the curve will pass through the coordinates and . We can sketch the graph shown below.

To solve the inequality , we consider the points on the graph of , where . This will be above the -axis at the values where is less than 0 and where is greater than 7. Because we do not have a strict inequality, can also be exactly equal to 0 or 7. Another way to express this would be by saying that is all the values excluding the points where is between 0 and 7. We can express this final answer in interval notation as

### Key Points

• We can solve quadratic inequalities by using a similar process to solving quadratic equations, by performing the same arithmetic operations to both sides of the inequality. However, we need to remember that when multiplying or dividing an inequality by a negative number, we must switch the inequality.
• A quadratic inequality has an interval of solutions, not just one distinct solution.
• To solve a quadratic inequality algebraically, we follow the steps below:
1. Rearrange the inequality so that we have all the terms of the expression on one side, with an inequality relating this to zero, for example, .
2. Factor the inequality by setting to identify the roots of the expression .
3. Identify the intervals that satisfy the inequality by using test points in each interval or a sign chart. We can also sketch a graph of the function.
• Take special care if you have rearranged the inequality to change the sign of the value; use the rearranged form of the inequality to identify the intervals, rather than the original inequality.
• To solve a quadratic inequality graphically, we follow the steps below.
1. Rearrange the inequality so that we have all the terms of the expression on one side, with an inequality relating this to zero; for example .
2. Factor the inequality by setting , to identify the roots of the expression .
3. Sketch the graph of the equation , using the roots of the equation and finding the direction of the parabola curve. Take special care if you have rearranged the original inequality to change the sign of the value: use the coefficient in the rearranged form of the inequality to identify the shape of the curve, rather than the original inequality.
4. Identify the intervals that satisfy the inequality.
• Our drawn graph does not need to be precise or even look particularly neat; it is simply there to give us a visual picture so that we can reach a conclusion about the intervals where the inequality will be true.