quadratic_inequalities

quadratic_inequalities - Solving quadratic inequalities A...

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Solving quadratic inequalities A sign chart of an expression is a number line that shows where the expression is positive, negative or 0. For example, the following picture shows a sign chart for - 2 x 3. We can construct the sign chart by first observing that = - 2 x 3 0 when = x 3 2 . Also - 2 x 3 > 0 when x > 3 2 and < - 2 x 3 0 when < x 3 2 . Example 1 : Solve the inequality: - x 2 x 2. Solution : - x 2 x 2 <=> - - x 2 x 2 0 <=> ( ) + x 1 ( ) - x 2 0. We can construct a sign chart for the factor ( ) + x 1 by first observing that = + x 1 0 when = x - 1. Also + x 1 > 0 when x > - 1 and < + x 1 0 when < x - 1. Similarly, we can construct a sign chart for the factor ( ) - x 2 by first observing that = - x 2 0 when = x 2. Also - x 2 > 0 when x > 2 and < - x 2 0 when < x 2. The product ( ) + x 1 ( ) - x 2 is positive when ( ) + x 1 and ( ) - x 2 have the same sign and negative when the two factors have opposite signs. Thus we obtain the following sign chart. Hence ( ) + x 1 ( ) - x 2 0 exactly when - 1 x 2. The solution set of the inequality is: [ ] , - 1 2 . Notes : (1) The last sign chart shows that = - - x 2 x 2 ( ) + x 1 ( ) - x 2 is positive when either x is greater than 2 or less than - 2. Hence - x 2 x > 2 <=> - - x 2 x 2 > 0 <=> < x - 1 or x > 2, and solution set of the "opposite" inequality - x 2 x > 2 is ( ) , -∞ - 1 ( ) , 2 .
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(2) The graph of = y - - x 2 x 2 is a parabola having the same shape as the graph of = y x 2 , but translated so that the vertex moves away from the origin. Since - - x 2 x 2 factors as ( ) + x 1 ( ) - x 2 , the x intercepts of the parabola occur where = x - 1 and = x 2. The line of symmetry is mid-way between these x intercepts and has equation = x 1 2 . The parabola opens upwards and so lies below the x axis between the x intercepts ( ) , - 1 0 and ( ) , 2 0 , and above the x axis when x is either greater than 2 or less than - 1. This indicates that the values of = y - - x 2 x 2 are less than 0 when x is between - 1 and 2 and greater than 0 when either < x - 1 or x > 2, that is, < - - x 2 x 2 0 when < - 1 x < 2 and - - x 2 x 2 > 0 when < x - 1 or x > 2. Example 2 : Solve the inequality: < - - 3 5 x 2 x 2 0. Solution : < - - 3 5 x 2 x 2 0 <=> + - 2 x 2 5 x 3 > 0 <=> ( ) - 2 x 1 ( ) + x 3 > 0. We can construct a sign chart for the factor ( ) - 2 x 1 by first observing that = - 2 x 1 0 when = x 1 2 . Also - 2 x 1 > 0 when x > 1 2 and < - 2 x 1 0 when < x 1 2 . Similarly, we can construct a sign chart for the factor ( ) + x 3 by first observing that = + x 3 0 when = x - 3. Also + x 3 > 0 when x > - 3 and < + x 3 0 when < x - 3. The product ( ) - 2 x 1 ( ) + x 3 is positive when ( ) - 2 x 1 and ( ) + x 2 have the same sign and negative when the two factors have opposite signs. Thus we obtain the following sign chart.
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Hence ( ) - 2 x 1 ( ) + x 3 > 0 exactly when < x - 3 or x > 1 2 . The solution set of the inequality is:
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quadratic_inequalities - Solving quadratic inequalities A...

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