Find the interval describing all solutions to the inequality 𝑥 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 𝑥 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, 𝑥 is equal to positive or negative two.
An alternative method would be to subtract four from both sides of the equation. This gives us 𝑥 squared minus four is equal to zero. We can then factor the quadratic into two sets of parentheses or brackets: 𝑥 plus two and 𝑥 minus two, as this is the difference of two squares. When 𝑥 plus two is equal to zero, 𝑥 is equal to negative two. When 𝑥 minus two is equal to zero, 𝑥 is equal to positive two or two. Our two solutions are 𝑥 equals negative two and 𝑥 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 𝑦 equals 𝑥 squared and see when it is equal to four, it will be easier to draw the graph 𝑦 equals 𝑥 squared minus four and see where it is equal to zero. Where our coefficient of 𝑥 squared is positive, we have a U-shaped parabola. If, on the other hand, we had a negative coefficient of 𝑥 squared, our parabola would be n-shaped.
The equation 𝑦 equals 𝑥 squared minus four has a 𝑦-intercept of negative four. It intersects the 𝑥-axis at 𝑥 equals two and 𝑥 equals negative two as these are the solutions of 𝑥 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 𝑥 squared is less than or equal to four. This is the same as the solutions of 𝑥 squared minus four is less than or equal to zero. This occurs when our curve is below the 𝑥-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 𝑛 values are included in the solution.
The interval that describes all the solutions to the inequality 𝑥 squared is less than or equal to four is negative two to two.