Graph x^2-4x-12< =0

Solution:

Step:1

Transform the given inequality into its equivalent equation form: $x^2-4x-12=0$.

Step:2

Apply the AC method to factorize the quadratic expression $x^2-4x-12$.

Step:2.1

Identify two numbers that multiply to give $c$ and add up to $b$, where $c=-12$ and $b=-4$. The numbers are $-6$ and $2$.

Step:2.2

Express the quadratic in its factored form using the identified numbers: $(x-6)(x+2)=0$.

Step:3

Recognize that if either factor equals $0$, the entire expression is $0$. Thus, set each factor to $0$: $x-6=0$ and $x+2=0$.

Step:4

Isolate $x$ in the equation $x-6=0$.

Step:4.1

Start with the equation $x-6=0$.

Step:4.2

Add $6$ to both sides to solve for $x$: $x=6$.

Step:5

Isolate $x$ in the equation $x+2=0$.

Step:5.1

Begin with the equation $x+2=0$.

Step:5.2

Subtract $2$ from both sides to solve for $x$: $x=-2$.

Step:6

The roots of the equation are the values that satisfy $(x-6)(x+2)=0$: $x=6$ and $x=-2$.

Step:7

Create test intervals based on the roots: $x<-2$, $-2<x<6$, and $x>6$.

Step:8

Select test values from each interval and substitute them into the original inequality to check for validity.

Step:8.1

For the interval $x<-2$, choose a test value and check the inequality.

Step:8.1.1

Select a test value, for example, $x=-4$.

Step:8.1.2

Substitute $x=-4$ into the inequality: $((-4{)}^2-4(-4)-12)\le 0$.

Step:8.1.3

Evaluate the inequality. The result is false as $20>0$.

Step:8.2

For the interval $-2<x<6$, choose a test value and check the inequality.

Step:8.2.1

Select a test value, for example, $x=0$.

Step:8.2.2

Substitute $x=0$ into the inequality: $((0{)}^2-4(0)-12)\le 0$.

Step:8.2.3

Evaluate the inequality. The result is true as $-12<0$.

Step:8.3

For the interval $x>6$, choose a test value and check the inequality.

Step:8.3.1

Select a test value, for example, $x=8$.

Step:8.3.2

Substitute $x=8$ into the inequality: $((8{)}^2-4(8)-12)\le 0$.

Step:8.3.3

Evaluate the inequality. The result is false as $20>0$.

Step:8.4

Determine which intervals satisfy the inequality. The interval $-2<x<6$ is true.

Step:9

The solution set includes all intervals that make the inequality true: $-2\le x\le 6$.

Step:10

Graph the solution on a number line, shading the interval from $-2$ to $6$ inclusively.

Knowledge Notes:

1. Quadratic Inequalities: A quadratic inequality is an inequality that involves a quadratic expression on one side of the inequality sign. It can be solved by finding the roots of the corresponding quadratic equation and then testing intervals to determine where the inequality holds true.

2. Factoring Quadratics: Factoring is a method used to express a quadratic expression as the product of two binomials. The AC method involves finding two numbers that multiply to give the product of the leading coefficient and the constant term (AC) and add up to the middle coefficient (B).

3. Testing Intervals: After finding the roots of the quadratic equation, the number line is divided into intervals. Test values from each interval are substituted into the original inequality to check which intervals satisfy the inequality.

4. Graphing Solutions: The solution to a quadratic inequality can be represented on a number line. Solid dots are used to indicate that a number is included in the solution set (鈮?or 鈮?, while open dots are used for numbers that are not included (< or >).

5. LaTeX Formatting: In the solution, mathematical expressions are formatted using LaTeX, a typesetting system that is widely used for mathematical and scientific documents. LaTeX allows for the clear presentation of mathematical equations and inequalities.