Question

solve the following inequality
$x^2 - 3x > 4$

select the correct choice below and, if necessary, fill in the answer box.
a. the solution set is \boxed{ } (type your answer in interval notation. use integers or fractions for any numbers in the expression.)
b. there is no real solution.

Explanation:

Step1: Rearrange the inequality

First, we rewrite the inequality \(x^{2}-3x > 4\) in standard quadratic form by subtracting 4 from both sides:
\(x^{2}-3x - 4>0\)

Step2: Factor the quadratic

We factor the quadratic expression \(x^{2}-3x - 4\). We need two numbers that multiply to - 4 and add to - 3. The numbers are - 4 and 1. So,
\(x^{2}-3x - 4=(x - 4)(x+1)\)
So the inequality becomes \((x - 4)(x + 1)>0\)

Step3: Find critical points

The critical points are the values of \(x\) that make the expression equal to zero. We set \((x - 4)(x + 1)=0\). Using the zero - product property, we get \(x-4 = 0\) or \(x + 1=0\). So the critical points are \(x = 4\) and \(x=-1\)

Step4: Test intervals

The critical points divide the number line into three intervals:

• Interval 1: \((-\infty,-1)\). We can test \(x=-2\). Substitute into \((x - 4)(x + 1)\): \((-2-4)(-2 + 1)=(-6)\times(-1)=6>0\). So the inequality is satisfied in this interval.
• Interval 2: \((-1,4)\). We can test \(x = 0\). Substitute into \((x - 4)(x + 1)\): \((0 - 4)(0+1)=(-4)\times1=-4<0\). So the inequality is not satisfied in this interval.
• Interval 3: \((4,\infty)\). We can test \(x = 5\). Substitute into \((x - 4)(x + 1)\): \((5 - 4)(5+1)=(1)\times(6)=6>0\). So the inequality is satisfied in this interval.

Answer:

The solution set is \((-\infty,-1)\cup(4,\infty)\)