Question

Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. (x2−x−2)/(x2−4x+3)\$$u003e0(x^2 -x$$ - $$2)/(x^2 - 4x + 3$$) \u003e 0

Accepted Answer

First, factor both the numerator and the denominator of the rational expression. For the numerator $$x^2 - x - 2$, find two numbers that multiply to -2$ and add to -1$. For the denominator x^2$ - 4x + 3$$, find two numbers that multiply to $$3$$ and add to $$-4. $$Rewrite the inequality using the factored forms: $$\frac{(x - a)(x - b)}{(x - c)(x - d)} \u003e 0$$, where $$a$$, $$b$$, $$c$$, and $$d$$ are the roots found from factoring. Identify the critical points by setting each factor equal to zero: $$x = a$$, $$x = b$$, $$x = c$$, and $$x = d. $$These points divide the real number line into intervals. Determine the sign of the expression on each interval by choosing a test point from each interval and substituting it into the factored inequality. Remember that the expression is undefined at points where the denominator is zero, so exclude those from the solution set. Based on the sign analysis, write the solution set where the expression is greater than zero. Then, graph this solution set on the real number line, marking excluded points (where the denominator is zero) with open circles.

Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. $(x^{2} - x - 2) / (x^{2} - 4 x + 3) > 0$

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Key Concepts

Rational Inequalities

Critical Points and Sign Analysis

Domain Restrictions and Excluded Values

Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. (x2−x−2)/(x2−4x+3)>0(x^2 -x - 2)/(x^2 - 4x + 3) > 0