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Solve the inequality 𝑥 minus five multiplied by 𝑥 minus seven is greater than or equal to negative five 𝑥 plus 35.
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In order to solve this inequality, we’ll begin by solving the equivalent equation, 𝑥 minus five multiplied by 𝑥 minus seven is equal to negative five 𝑥 plus 35.
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Distributing the parentheses or expanding the brackets using the FOIL method on the left-hand side gives us 𝑥 squared minus seven 𝑥 minus five 𝑥 plus 35.
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We notice here that we have negative five 𝑥 and positive 35 on both sides of the equation.
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By adding five 𝑥 and subtracting 35 from both sides, these terms will cancel.
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This leaves us with the equation 𝑥 squared minus seven 𝑥 is equal to zero.
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Factoring out the highest common factor of 𝑥 gives us 𝑥 multiplied by 𝑥 minus seven is equal to zero.
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This gives us two solutions: 𝑥 equals zero or 𝑥 equals seven.
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Solving the inequality 𝑥 minus five multiplied by 𝑥 minus seven is greater than or equal to negative five 𝑥 plus 35 is the same as solving the inequality 𝑥 squared minus seven 𝑥 is greater than or equal to zero.
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When the coefficient of 𝑥 squared in any quadratic equation is positive, we have a U-shaped parabola.
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When we have a negative coefficient of 𝑥 squared, we have an n-shaped parabola.
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The equation 𝑦 equals 𝑥 squared minus seven 𝑥 has a 𝑦-intercept of zero and crosses the 𝑥-axis at zero and seven.
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We are looking for the values where this is greater than or equal to zero.
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This is the points above or on the 𝑥-axis.
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This is true for all 𝑥-values less than or equal to zero or greater than or equal to seven.
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We could write this as the set of values from negative ∞ to zero, including zero, or the set of values from seven up to ∞.
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Note that the square brackets mean that we include zero and seven.
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An alternative way of writing this would be all of the real values except those between zero and seven.
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The solutions of the inequality 𝑥 minus five multiplied by 𝑥 minus seven that are greater than or equal to negative five 𝑥 plus 35 are all the real values except those between zero and seven.
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We could check this answer by substituting in values to the original inequality.