Video: Solving Quadratic Inequalities

Solve the inequality (π‘₯ βˆ’ 5)(π‘₯ βˆ’ 7) β‰₯ βˆ’5 + 35.

03:15

Video Transcript

Solve the inequality π‘₯ minus five multiplied by π‘₯ minus seven is greater than or equal to negative five π‘₯ plus 35.

In order to solve this inequality, we’ll begin by solving the equivalent equation, π‘₯ minus five multiplied by π‘₯ minus seven is equal to negative five π‘₯ plus 35. Distributing the parentheses or expanding the brackets using the FOIL method on the left-hand side gives us π‘₯ squared minus seven π‘₯ minus five π‘₯ plus 35.

We notice here that we have negative five π‘₯ and positive 35 on both sides of the equation. By adding five π‘₯ and subtracting 35 from both sides, these terms will cancel. This leaves us with the equation π‘₯ squared minus seven π‘₯ is equal to zero. Factoring out the highest common factor of π‘₯ gives us π‘₯ multiplied by π‘₯ minus seven is equal to zero. This gives us two solutions: π‘₯ equals zero or π‘₯ equals seven.

Solving the inequality π‘₯ minus five multiplied by π‘₯ minus seven is greater than or equal to negative five π‘₯ plus 35 is the same as solving the inequality π‘₯ squared minus seven π‘₯ is greater than or equal to zero.

When the coefficient of π‘₯ squared in any quadratic equation is positive, we have a U-shaped parabola. When we have a negative coefficient of π‘₯ squared, we have an n-shaped parabola. The equation 𝑦 equals π‘₯ squared minus seven π‘₯ has a 𝑦-intercept of zero and crosses the π‘₯-axis at zero and seven.

We are looking for the values where this is greater than or equal to zero. This is the points above or on the π‘₯-axis. This is true for all π‘₯-values less than or equal to zero or greater than or equal to seven. We could write this as the set of values from negative ∞ to zero, including zero, or the set of values from seven up to ∞. Note that the square brackets mean that we include zero and seven. An alternative way of writing this would be all of the real values except those between zero and seven.

The solutions of the inequality π‘₯ minus five multiplied by π‘₯ minus seven that are greater than or equal to negative five π‘₯ plus 35 are all the real values except those between zero and seven. We could check this answer by substituting in values to the original inequality.

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