Question

solve for x and graph the solution.
$x^2 - 3x - 4 \leq 0$
plot the endpoints. select an endpoint to change it from closed to open. select the middle of a segment, ray, or line to delete it.
<---|-6-|-5-|-4-|-3-|-2-|-1-|0-|1-|2-|3-|4-|5-|6--->
submit

Explanation:

Step1: Factor the quadratic expression

Factor \(x^{2}-3x - 4\). We look for two numbers \(a\) and \(b\) such that \(a + b=-3\) and \(ab=-4\). The numbers are \(a = 1\) and \(b=-4\). So, \(x^{2}-3x - 4=(x + 1)(x - 4)\). The inequality becomes \((x + 1)(x - 4)\leq0\).

Step2: Find the critical points

Set \((x + 1)(x - 4)=0\). Using the zero - product property \(x+1 = 0\) gives \(x=-1\), and \(x - 4=0\) gives \(x = 4\). These are the critical points that divide the number line into three intervals: \((-\infty,-1)\), \((-1,4)\), and \((4,\infty)\).

Step3: Test the intervals

• For the interval \((-\infty,-1)\), let \(x=-2\). Then \((-2 + 1)(-2 - 4)=(-1)\times(-6)=6>0\).
• For the interval \((-1,4)\), let \(x = 0\). Then \((0 + 1)(0 - 4)=(1)\times(-4)=-4<0\).
• For the interval \((4,\infty)\), let \(x=5\). Then \((5 + 1)(5 - 4)=(6)\times(1)=6>0\).

Step4: Consider the equality case

Since the inequality is \(\leq\), the endpoints \(x=-1\) and \(x = 4\) are included in the solution set.

Answer:

The solution of the inequality \(x^{2}-3x - 4\leq0\) is \([-1,4]\). On the number - line, we plot closed circles at \(x=-1\) and \(x = 4\) and shade the segment between them.