Inequality Netation:

Number Line:

Interval Notation:

To solve the inequality , we start by finding the roots of the quadratic equation using the quadratic formula . Here, . The discriminant, , is positive, indicating two real roots. Plugging into the formula gives us , leading to roots at and . To find the intervals where the inequality holds, we can test points in the intervals , , and .

The expression is negative in the interval .

So the solutions are:

The quadratic can be graphed as a parabola opening upwards. The roots indicate where the curve crosses the x-axis, meaning that between these roots (i.e., to ), the function takes negative values. Always a good visual cue – if you see the parabola below the x-axis, you know you’re in the solution range!

When solving inequalities involving quadratics, a common mistake is to forget about the direction of the inequality when checking test points. Always remember to pick a test point from each interval created by the roots to verify whether it satisfies the inequality. Don’t rush through those test points – they can make all the difference!