Question

How do you graph the solution to \[\left( {x + 4} \right)\left( {x - 6} \right) > 0\] on a number line?

Answer 1

Hint: In these types of questions as the given inequalities should be factored and the given factors should be tested in the ranges and the intervals should satisfy the given inequality and finally the intervals should be plotted on the graph.

Complete step-by-step answer:
Given inequality is \[\left( {x + 4} \right)\left( {x - 6} \right) > 0\],
Now we will multiply both the terms and equate the result to zero to find the critical point of x,
\[\left( {x + 4} \right)\left( {x - 6} \right) > 0\]
Now multiplying and equate it 0, we get,
\[{x^2} + 4x - 6x - 24 = 0\],
Now factorising we get,
\[{x^2} + 2x - 24 = 0\],
So, now we get,
\[\left( {x + 4} \right)\left( {x - 6} \right) = 0\],
Now we get the critical points of x we get,
\[x + 4 = 0\], or \[x - 6 = 0\],
Now set factors to zero we get,
\[x = - 4\], and
\[x = 6\],
Now check the intervals in between the critical points, i.e., test the intervals to see if they satisfy the given inequality.
Let’s check \[x < - 4\], let us take any value in the range, let us take \[x = - 5\], we get
\[\left( { - 5 + 4} \right)\left( { - 5 - 6} \right) > 0\],
By simplifying we get,
\[\left( { - 1} \right)\left( { - 11} \right) > 0\],
By multiplying we get,
\[11 > 0\], satisfy the given inequality,
Now let us check the range \[ - 4 < x < 6\], now we will take a value between the ranges let us take 2, we will get,
\[\left( {2 + 4} \right)\left( {2 - 6} \right) > 0\],
Now multiplying we get,
\[\left( 6 \right)\left( { - 4} \right) > 0\],
Again by simplifying we get,
\[ - 24 > 0\], which doesn’t satisfy the given inequality.
Now let us check the range \[x > 6\], let us take a value In the range, so let us take 7, we will get,
\[\left( {x + 4} \right)\left( {x - 6} \right) > 0\],
Now substitute \[x = 7\] in the equality we get,
\[\left( {7 + 4} \right)\left( {7 - 6} \right) > 0\],
Now multiplying we get,
\[\left( {11} \right)\left( 1 \right) > 0\],
Now again simplifying we get,
\[11 > 0\], which satisfies the given inequality.
So the required solution for the given inequality is \[x < - 4\] or \[x > 6\].
Now plotting the solution on the graph we get,

Final Answer:
The required graph of the given inequality \[\left( {x + 4} \right)\left( {x - 6} \right) > 0\] is,

Note:
While solving these types of questions we should be careful in calculations as there is a chance of making a mistake in the calculations, and should plot the intervals on the number line.