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Chapter 5: Problem 62

Solve and graph the solution set on a number line: $$ x^{2}-4 x>-3 $$

Short Answer

Expert verified
The solution to the inequality \(x^{2}-4x > -3\) is \((-\infty,1) \cup (3,+\infty)\).

Step by step solution

01

- Convert Inequality into Equality

Consider the given inequality, \(x^{2}-4x > -3\). In order to solve it, first convert the inequality into equality. So, \(x^{2}-4x = -3\).
02

- Bring All Terms to One Side

Subtract -3 from both sides to make the right side 0. Then equation becomes \(x^{2}-4x + 3 = 0\).
03

- Solve the Equations for Roots

Solve the equation \(x^{2}-4x + 3 = 0\) to find the roots. This can be factorized into \((x-3)(x-1)=0\). Therefore, the roots of the equation are x = 3 and x = 1.
04

- Test the Inequality within the intervals

Now, we create intervals based on the roots. These intervals are \((-\infty,1]\), \((1,3]\), and \((3,+\infty)\). Select any number within the intervals and test the original inequality.
05

- Draw the Number Line and Plot the Solution

The solution intervals are those that satisfy the original inequality. Draw the number line and plot the solutions. Only those intervals on the number line should be considered in the final solution, which satisfy the original inequality.

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