Question

solve the quadratic inequality (x-2)(x+4)>0 using desmos. select your answer. a. $x < -4$ or $x > 2$ b. $-4 < x < 2$ c. $x > -4$ and $x < 2$ d. $x = -4$ or $x = 2$ select test points to verify your solution set. for $x < $ __ if $x = $ for $< x < $ if $x = $ for $x > $ if $x = $ therefore, the solution set is ______.

Explanation:

Part 1: Solve the Quadratic Inequality

Step 1: Find the roots of the equation \((x - 2)(x + 4)=0\)

To find the roots, we set each factor equal to zero:

• For \(x - 2 = 0\), we get \(x = 2\).
• For \(x + 4 = 0\), we get \(x = -4\).

These roots divide the number line into three intervals: \(x < -4\), \(-4 < x < 2\), and \(x > 2\).

Step 2: Analyze the sign of \((x - 2)(x + 4)\) in each interval

• Interval 1: \(x < -4\)

Let's take a test point, say \(x = -5\).
Substitute into \((x - 2)(x + 4)\):
\((-5 - 2)(-5 + 4)=(-7)(-1)=7>0\).
So, the product is positive in this interval.

• Interval 2: \(-4 < x < 2\)

Let's take a test point, say \(x = 0\).
Substitute into \((x - 2)(x + 4)\):
\((0 - 2)(0 + 4)=(-2)(4)=-8<0\).
So, the product is negative in this interval.

• Interval 3: \(x > 2\)

Let's take a test point, say \(x = 3\).
Substitute into \((x - 2)(x + 4)\):
\((3 - 2)(3 + 4)=(1)(7)=7>0\).
So, the product is positive in this interval.

We need to find where \((x - 2)(x + 4)>0\). From the analysis, this happens when \(x < -4\) or \(x > 2\).

We use the intervals and test points from the previous analysis to fill in the table.

For \(x < -4\):

• The boundary is \(-4\), so we write \(x < -4\).
• A test point can be \(x = -5\) (as used earlier).

For \(-4 < x < 2\):

• The boundaries are \(-4\) and \(2\), so we write \(-4 < x < 2\).
• A test point can be \(x = 0\) (as used earlier).

For \(x > 2\):

• The boundary is \(2\), so we write \(x > 2\).
• A test point can be \(x = 3\) (as used earlier).

Solution Set:

From the sign analysis, the solution set is \(x < -4\) or \(x > 2\).

Answer:

(for the multiple-choice part):
A. \(x < -4\) or \(x > 2\)

Part 2: Verify the Solution Set with Test Points