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Chapter 2: Problem 6

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$x^{2}-4 x+3<0$$

Short Answer

Expert verified
The solution set for \(x^{2}-4x+3 < 0\) on the real number line is \((1, 3)\)

Step by step solution

01

Factorize the polynomial

The polynomial \(x^{2}-4x+3\) can be factored as \((x - 3)(x - 1)\)
02

Solve the equation

Solving the equation \((x - 3)(x - 1) = 0\), solutions are \(x = 1\) and \(x = 3\). These are the critical points.
03

Sign chart test

Choose any test points in the intervals \(-\infty, 1\), \(1, 3\), and \(3, \infty\) and substitute into \((x - 3)(x - 1)\). If the resulting value is less than zero, the interval satisfies the inequality. The results are: For \(x < 1\), \((x - 3)(x - 1) > 0\), indicating '+'; For \(1 < x < 3\), \((x - 3)(x - 1) < 0\), indicating '-'; For \(x > 3\), \((x - 3)(x - 1) > 0\), indicating '+'
04

Write interval notation

Since only negative values satisfy the inequality, the solution in interval notation is \((1, 3)\)

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