Quadratic inequalities have the form *ax*²+*bx*+*c*<0. Inequalities can use the greater than, less than, greater than or equal to, and less than or equal to signs. To solve these types of inequalities, we have to determine the points where the graph of the quadratic function will intersect the *x*-axis. Sketching a simple graph is always helpful.

Here, we will learn how to solve quadratic inequalities step-by-step. In addition, we will look at some examples with answers to learn the concepts.

## Steps to solve quadratic inequalities

To solve quadratic inequalities, we can follow the following steps:

**Step 1:** Simplify and write the inequality in the form $latex ax^2+bx+c<0$. The “<” sign could be different depending on the problem.

**Step 2:** Identify where the graph of $latex y=ax^2+bx+c$ intersects the *x*-axis. To achieve this, we can solve the quadratic equation by factoring $latex ax^2+bx+c=0$ and find the *x* values.

**Step 3:** Sketch a simple graph of the function $latex y=ax^2+bx+c$ to determine the solution. Alternatively, we can solve without a graph by considering the following:

- If we have a positive quadratic term, the parabola opens up and is U-shaped.
- If the quadratic term is negative, the parabola opens down.
- Values below the
*x*-axis are less than 0, and values above the*x*-axis are greater than 0.

**Step 4:** Using the graph or otherwise, we need to determine the inequality symbols that will make the solutions found in step 2 satisfy the inequality.

Explore the examples with answers shown below to understand the application of these steps with real problems.

## Quadratic inequalities – Examples with answers

The steps of solving quadratic inequalities are applied to solve the following examples. Try to solve the problems yourself before looking at the solution.

**EXAMPLE 1**

Solve the inequality $latex x^2+3x-4<0$.

##### Solution

**Step 1:** The inequality is already in the form $latex ax^2+bx+c<0$.

**Step 2:** We form the equation $latex x^2+3x-4=0$ and find the values of *x*. We can solve this by factoring:

$latex x^2+3x-4=0$

$latex (x+4)(x-1)=0$

$latex x=-4~~$ or $latex ~~x=1$

**Step 3:** The quadratic term is positive, so the parabola opens up:

**Step 4:** The inequality $latex x^2+3x-4<0$ tells us that the expression $latex y=x^2+3x-4$ is less than 0. Looking at the graph, we can conclude that this happens when $latex-4<x<1$.

That is, the values on the graph are negative (below the *x*-axis) when the values of *x* range from -4 to 1.

**EXAMPLE **2

**EXAMPLE**

Find the solution to the inequality $latex x^2+x-2>0$.

##### Solution

**Step 1:** The inequality is already in the form $latex ax^2+bx+c<0$.

**Step 2:** We are going to use factorization to solve the equation $latex x^2+x-2=0$:

$latex x^2+x-2=0$

$latex (x+2)(x-1)=0$

$latex x=-2~~$ or $latex ~~x=1$

**Step 3:** The parabola is U-shaped and opens up since the quadratic term is positive:

**Step 4:** The inequality $latex x^2+x-2>0$ tells us that the expression $latex y=x^2+x-2$ is greater than 0. Using the graph, we can conclude that this happens when $ latex x<-2~$ or $latex ~x>1$.

That is, the values of the graph are positive (above the *x*-axis) when the values of *x* are less than -2 and greater than 1.

**EXAMPLE **3

**EXAMPLE**

Find the solution to the inequality $latex x^2+2x-4<x+2$.

##### Solution

**Step 1:** We write the inequality in the form $latex ax^2+bx+c<0$:

$latex x^2+2x-4<x+2$

$latex x^2+x-6<0$

**Step 2:** Solving the equation $latex x^2+x-6=0$, we have:

$latex x^2+x-6=0$

$latex (x+3)(x-2)=0$

$latex x=-3~~$ or $latex ~~x=2$

**Step 3:** Since the quadratic term is positive, the parabola opens up.

**Step 4:** In this case, we have that the expression $latex y=x^2+x-6$ is less than 0. Thus, we need the part of the parabola that is below the *x*-axis.

Since the parabola opens up, the parabola is below the *x*-axis when the values of *x* are greater than -3 and less than 2, that is, $latex -3<x<2$.

**EXAMPLE **4

**EXAMPLE**

What is the solution to the inequality $latex x^2+8x+4>2x-4$?

##### Solution

**Step 1:** We simplify the inequality as follows:

$latex x^2+8x+4>2x-4$

$latex x^2+6x+8>0$

**Step 2:** Solving the equation $latex x^2+6x+8=0$, we have:

$latex x^2+6x+8=0$

$latex (x+4)(x+2)=0$

$latex x=-4~~$ or $latex ~~x=-2$

**Step 3:** The quadratic term is positive. Therefore, the parabola is U-shaped and opens up.

**Step 4:** The inequality $latex x^2+6x+8>0$ tells us that the expression $latex y=x^2+6x+8$ is greater than 0. That is, we need the part of the parabola that is above the x-axis.

Since the parabola opens up, the solution to the inequality is $latex x<-4~$ or $latex ~x>-2$.

**EXAMPLE **5

**EXAMPLE**

Solve the inequality $latex -x^2-x+6>0$.

##### Solution

**Step 1:** In this case, we have an inequality with a negative quadratic term. To facilitate its resolution, we can multiply the entire inequality by -1:

$latex x^2+x-6<0$

**Note:** Remember that when we multiply or divide the inequality by a negative sign, we have to flip the inequality sign.

**Step 2:** Solving the equation $latex x^2+x-6=0$ by factoring, we have:

$latex x^2+x-6=0$

$latex (x+3)(x-2)=0$

$latex x=-3~~$ or $latex ~~x=2$

**Step 3:** The parabola formed by $latex y=x^2+x-6$ opens up because the quadratic term is positive.

**Step 4:** We need the negative values of the parabola because we have the quadratic expression to be less than 0. Then, we look for the portion of the parabola below the *x*-axis.

Therefore, the solution is $latex -3<x<2$.

**EXAMPLE **6

**EXAMPLE**

Solve the inequality $latex \frac{x+1}{7x-1}\leq \frac{2}{7}$.

##### Solution

**Step 1:** We can multiply the entire inequality by $latex (7x-1)^2$ to simplify and make sure the inequality is positive. Therefore, we have:

$latex (x+1)(7x-1)\leq \frac{2}{7}(7x-1)^2$

$latex 7x^2+6x-1\leq \frac{2}{7}(49x^2-14x+1)$

$latex 0\leq 7x^2-10x+\frac{9}{7}$

$latex 49x^2-70x+9\geq 0$

**Step 2:** We can solve the equation $latex 49x^2-70x+9=0$ by factoring. Thus, we have:

$latex 49x^2-70x+9=0$

$latex (7x-1)(7x-9)=0$

$latex x=\frac{1}{7}~~$ or $latex ~~x=\frac{9}{7}$

**Step 3:** The parabola opens up and is U-shaped because the quadratic term is positive.

**Step 4:** The inequality $latex 49x^2-70x+9\geq 0$ tells us that the expression $latex y=49x^2-70x+9$ is greater than or equal to 0. Therefore, we need the part of the parabola that is above the *x*-axis, including the points that cut the *x*-axis.

Since the parabola opens up, the solution is $latex x\leq \frac{1}{7}~$ or $latex ~x\geq \frac{9}{7}$.

## Quadratic inequalities – Practice problems

Find the solution to the following practice problems to apply everything learned about quadratic inequalities.

## See also

Interested in learning more about quadratic inequalities? You can take a look at these pages:

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