# Quadratic Inequalities – Explanation and Examples

Quadratic inequalities are quadratic expressions that use inequality signs. These types of inequalities have the general form ax²+bx+c<0, where the inequality sign can be <, >, ≤, or ≥. The solution to a quadratic inequality can be found using its graph.

In this article, we will learn all about quadratic inequalities. We will learn how to solve them, and we will apply what we have learned in some practice problems.

Relevant for

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Relevant for

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## What are quadratic inequalities?

Quadratic inequalities are quadratic expressions that use inequality signs to compare two quantities. The following are examples of quadratic inequalities:

• $latex 3x^2+2x-6<0$
• $latex 2x^2-5x+3>0$
• $latex -2x^2-6x+4\leq 0$
• $latex 5x^2+4x-5\geq 0$

## How to solve quadratic inequalities?

We can solve quadratic inequalities by following these steps:

Step 1: If it isn’t, write the inequality in the form $latex ax^2+bx+c>0$.

Note: The “>” sign will depend on the problem.

Step 2: Find the values of x such that $latex ax^2+bx+c=0$. These are the points where the function $latex y=ax^2+bx+c$ intersects the x-axis. We can solve this by factoring the quadratic expression.

Step 3: Obtain a simple graph of the function $latex y=ax^2+bx+c$, or alternatively, consider the following:

• When the quadratic term is positive, the parabola opens up and is U-shaped.
• When the quadratic term is negative, the parabola opens down.
• Values above the x-axis are greater than 0, and values below the x-axis are less than 0.

Step 4: Determine the inequality symbols that will make the solutions found in step 2 satisfy the inequality. We can accomplish this by using the graph or the statements from step 3.

Look at the following examples, where we apply this process to solve quadratic inequalities.

## 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

Find the solution to the inequality $latex x^2+3x-4<0$.

Step 1: We don’t have to simplify because the inequality is already in the form $latex ax^2+bx+c<0$.

Step 2: We can form the equation $latex x^2+3x-4=0$ and find the values of x using factorization:

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

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

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

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

Step 4: The inequality $latex x^2+3x-4<0$ tells us that we need the part that produces negative values in the parabola. Using the graph, we can deduce that this happens when $latex -4<x<1$.

This means that the values of the parabola are negative (below the x-axis) when the values of x range from -4 to 1.

### EXAMPLE 2

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

Step 1: We have nothing to simplify.

Step 2: We can solve the equation $latex x^2+x-2=0$ using factorization:

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

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

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

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

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

That means 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

Solve the inequality $latex x^2+2x-12<x+8$.

Step 1: We can simplify by combining like terms and writing the inequality in the form $latex ax^2+bx+c<0$:

$latex x^2+2x-12<x+8$

$latex x^2+x-20<0$

Step 2: We form the equation $latex x^2+x-20=0$ and solve it using factorization:

$latex x^2+x-20=0$

$latex (x+5)(x-4)=0$

$latex x=-5~~$ or $latex ~~x=4$

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

Step 4: We have the inequality $latex x^2+x-20<0$, which means we need the portion that is less than 0. Therefore, we look for 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 -5 and less than 4, that is, $latex -5<x<4$.

### EXAMPLE 4

Find the solution to the inequality $latex x^2+4x+10>-4x-5$.

Step 1: We can combine like terms and write the inequality as follows:

$latex x^2+4x+10>-4x-5$

$latex x^2+8x+15>0$

Step 2: Solving the equation $latex x^2+8x+15=0$ by factorization, we have:

$latex x^2+8x+15=0$

$latex (x+5)(x+3)=0$

$latex x=-5~~$ or $latex ~~x=-3$

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

Step 4: The inequality $latex x^2+8x+15>0$ tells us that we need the part that is greater than 0, that is, the part of the parabola that is above the x-axis.

Since the parabola opens up, the solution to the inequality is $latex x<-5~$ or $latex ~x>-3$.

### EXAMPLE 5

Find the solution to the inequality $latex -x^2+6x-8>0$.

Step 1: In this case, the quadratic term of the inequality is negative. To facilitate its resolution, we can multiply the entire inequality by -1:

$latex x^2-6x+8<0$

Note: When we multiply or divide the inequality by a negative sign, we have to flip the inequality sign.

Step 2: We can form the equation $latex x^-6x+8=0$ and solve by factoring:

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

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

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

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

Step 4: The inequality $latex x^2-6x+8<0$ indicates that we need the part that is less than 0. Therefore, we look for the portion of the parabola below the x-axis.

The solution is $latex 2<x<4$.

### EXAMPLE 6

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

Step 1: Let’s multiply the entire inequality by $latex (7x-1)^2$ to simplify and make sure the inequality is positive:

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

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

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

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

Step 2: Forming the equation $latex 49x^2-70x+9=0$ and solving by factoring, we find the x-intercepts:

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

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

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

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

Step 4: The inequality $latex 49x^2-70x+9\geq 0$ tells us that we need to find the part that is greater than or equal to 0. Thus, we use the part of the parabola that is above the x-axis, including the points that intersect 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.

#### Find the solution to the inequality $latex x^2+5x\geq 10+2x$.  