# Question Video: Finding the Solution Set of an Inequality of the Second Degree Mathematics • 10th Grade

Find the interval describing all solutions to the inequality π₯Β² β€ 4.

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### Video Transcript

Find the interval describing all solutions to the inequality π₯ squared is less than or equal to four.

In order to solve any quadratic inequality, we begin by solving the equivalent equation. In this case, we solve π₯ squared is equal to four. One way to solve this is to square root both sides of the equation. As the square root of four is equal to two, π₯ is equal to positive or negative two.

An alternative method would be to subtract four from both sides of the equation. This gives us π₯ squared minus four is equal to zero. We can then factor the quadratic into two sets of parentheses or brackets: π₯ plus two and π₯ minus two, as this is the difference of two squares. When π₯ plus two is equal to zero, π₯ is equal to negative two. When π₯ minus two is equal to zero, π₯ is equal to positive two or two. Our two solutions are π₯ equals negative two and π₯ equals positive two.

In order to find all the solutions to the inequality, it then helps to draw a graph of the function. Whilst we could draw the graph of the equation π¦ equals π₯ squared and see when it is equal to four, it will be easier to draw the graph π¦ equals π₯ squared minus four and see where it is equal to zero. Where our coefficient of π₯ squared is positive, we have a U-shaped parabola. If, on the other hand, we had a negative coefficient of π₯ squared, our parabola would be n-shaped.

The equation π¦ equals π₯ squared minus four has a π¦-intercept of negative four. It intersects the π₯-axis at π₯ equals two and π₯ equals negative two as these are the solutions of π₯ squared minus four is equal to zero. Drawing a smooth curve through these points gives us the following graph.

In the question, we wanted the points where π₯ squared is less than or equal to four. This is the same as the solutions of π₯ squared minus four is less than or equal to zero. This occurs when our curve is below the π₯-axis. This is all the values between negative two and two inclusive. As our inequality sign was less than or equal to, we use square brackets. This means that these π values are included in the solution.

The interval that describes all the solutions to the inequality π₯ squared is less than or equal to four is negative two to two.