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What are the values of π₯ for which the functions π of π₯ is equal to π₯ minus five and π of π₯ is equal to π₯ squared plus two π₯ minus 48 are both positive?
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Letβs begin by considering the function π of π₯ is equal to π₯ minus five.
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If we want this to be positive, π of π₯ must be greater than zero.
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This gives us π₯ minus five is greater than zero.
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Adding five to both sides of this inequality gives us π₯ is greater than five.
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π of π₯ is therefore positive on the open interval five to β.
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It is a positive function for any value greater than five.
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We will now repeat this process for π of π₯.
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This gives us π₯ squared plus two π₯ minus 48 is greater than zero.
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To solve any quadratic inequality of this form, we firstly need to find the zeros by setting our function equal to zero.
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π₯ squared plus two π₯ minus 48 equals zero.
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This can be factored or factorized into two sets of parentheses or brackets.
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The first term in each bracket is π₯.
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The second terms need to have a product of negative 48 and a sum of two.
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Six multiplied by eight is equal to 48.
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This means that negative six multiplied by eight is equal to negative 48.
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Negative six plus eight is equal to two.
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Our two sets of parentheses are π₯ minus six and π₯ plus eight.
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As the product of these two terms is equal to zero, either π₯ minus six equals zero or π₯ plus eight is equal to zero.
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Adding six to both sides of the first equation gives us π₯ is equal to six.
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And subtracting eight from both sides of the second equation gives us π₯ is equal to negative eight.
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This means that the function π of π₯ is equal to zero when π₯ equals six and π₯ equals negative eight.
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As our function is quadratic and the coefficient of π₯ squared is positive, the graph will be u-shaped.
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This means that it is positive on two sections, when π₯ is greater than six and when π₯ is less than negative eight.
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The solution to the inequality π₯ squared plus two π₯ minus 48 is greater than nought is π₯ is less than negative eight or π₯ is greater than six.
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This can also be written using interval notation.
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π of π₯ is positive in the open interval negative β to negative eight or the open interval six to β.
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We want to work out the values of π₯ where both functions are positive.
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Letβs consider a number line with the key values five, negative eight, and six marked on.
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We know that π of π₯ is positive for all values greater than five.
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π of π₯ is positive for all values less than negative eight and greater than six.
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This means that both functions are positive when π₯ is greater than six.
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This could also be written using interval notation as the open interval six to β.