Quadratic Inequalities - Examples and Practice

To solve the inequality, we need to find the roots of the equation $latex x^2+3x-4=0$. We can achieve this using factorization:

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

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

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

Now, we can sketch a graph to facilitate the resolution of the problem:

We have the inequality $latex x^2+3x-4<0$, which tells us that we need the part that has values less than 0. That is, we need the part that is below the x-axis.

Using the graph, we can deduce that this happens when $latex -4<x<1$.

To solve the inequality, we need to start by finding the values of x in the equation $latex x^2+x-2=0$. Therefore, we can use factorization:

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

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

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

The parabola formed by the equation $latex y=x^2+x-2$ opens up and is U-shaped because the quadratic term is positive:

The inequality $latex x^2+x-2>0$ indicates that the values are always greater than 0. Using the graph, we can deduce that this happens when $latex x<-2~$ or $latex ~x>1$.

We need to find the values of x in the equation $latex x^2+2x-8=0$. We can achieve this using factorization:

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

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

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

Let's solve without using the graph. For this, we have to consider that since the quadratic term is positive, the parabola formed by $latex y=x^2+2x-8$ will open up.

Now, the inequality $latex x^2+2x-8<0$ indicates that we need only the values that are less than 0, that is, the part that is below the x-axis. This happens when $latex -4<x<2$.

We start by simplifying as follows:

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

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

Now, we can form the equation $latex x^2+6x+8=0$ and solve by factoring:

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

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

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

The inequality $latex x^2+6x+8>0$ means that the values are always greater than 0. This means that we need the part of the parabola that is above the x-axis.

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

We can combine like terms and write the inequality in the form $latex ax^2+bx+c<0$:

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

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

We can find the values of x by forming the equation $latex x^2+x-20=0$ and solve by using factorization:

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

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

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

Now, we need to find the values of x that make $latex x^2+x-20<0$ always have values less than 0. This means we need the part below the x-axis.

Since the parabola opens up (the quadratic term is positive), the parabola is below the x-axis when the values of x are greater than -5 and less than 4, that is, $latex -5<x<4$.

Simplifying the inequality, we get the following:

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

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

Now, we can form the equation $latex x^2+8x+15=0$ and solve it by factoring:

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

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

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

To solve the inequality $latex x^2+8x+15>0$, we have to find the values of x that always produce values greater than 0 (above the x-axis).

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

Here, we have a quadratic term that is negative. We can make it easier to solve the inequality by multiplying the entire inequality by -1 to make the quadratic term positive:

$latex x^2-6x+8\leq 0$

Note: Keep in mind that when multiplying or dividing the inequality by a negative sign, we have to flip the inequality sign.

Now, we form the equation $latex x^-6x+8=0$ and solve by factoring to find the values of x:

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

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

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

To solve the inequality $latex x^2-6x+8\leq 0$, we have to look for the values of x that produce values less than or equal to 0. Therefore, we look for the portion of the parabola below the x-axis.

The solution is $latex 2\leq x\leq 4$.

Again, we can multiply the entire inequality by -1 to make it easier to solve:

$latex 2x^2-9x+10\geq 0$

Note: The inequality sign was changed since we multiplied by a negative sign.

Now, we solve the equation $latex 2x^-9x+10=0$:

$latex 2x^2-9x+10=0$

$latex (2x-5)(x-2)=0$

$latex x=\frac{5}{2}~~$ or $latex ~~x=2$

The inequality $latex 2x^2-9x+10\geq 0$ indicates that we need the values of x that produce values greater than or equal to 0. Thus, we look for the portion of the parabola above the x-axis.

The solution is $latex x\leq 2~~$ or $latex ~~x\geq \frac{5}{2}$.

Let's multiply the entire inequality by $latex (x-3)^2$ to get rid of the fraction and make sure the inequality is positive

$latex 2(x-3)\leq 1(x-3)^2$

$latex 2x-6\leq x^2-6x+9$

$latex 0\leq x^2-8x+15$

$latex x^2-8x+15\geq 0$

We solve the equation $latex x^2-8x+15=0$ by factoring:

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

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

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

Now, we need to find the values of x that produce values greater than or equal to 0.

Since the parabola opens up, the solution is $latex x\leq 3~$ or $latex ~x\geq 5$.

We can multiply the entire inequality by $latex (7x-1)^2$ to get rid of the fraction and make sure the inequality is positive

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

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

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

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

Now, we form the equation $latex 49x^2-70x+9=0$ and solve by factoring:

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

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

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

To solve the inequality $latex 49x^2-70x+9\geq 0$, we need to find the values of x that produce values greater than or equal to 0.

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