Solve the inequality 𝑥 squared minus three 𝑥 is less than or equal to 10.
To solve a quadratic inequality, we treat it like we would a quadratic equation and
we rearrange it so that we’ve got zero on one side. In this case, we’ll subtract 10 from both sides, which gives us 𝑥 squared minus
three 𝑥 minus 10 is less than or equal to zero.
Now, we replace the inequality symbol with an equal sign. We’re going to solve this quadratic equation by factorizing the expression 𝑥 squared
minus three 𝑥 minus 10. To do this, we find two numbers that multiply to make negative 10 and add to make
negative three. The factor pairs of negative 10 are negative one and 10, one and negative 10,
negative two and five, and two and negative five. The pair of numbers that multiply to make negative 10 and add to make negative three
are two and negative five. That gives us a factorized expression of 𝑥 plus two multiplied by 𝑥 minus five.
We know that a single 𝑥 must go at the beginning of the brackets because when we
multiply them back out, 𝑥 multiplied by 𝑥 is 𝑥 squared, which is what we needed
in our expression. We said that our expression is equal to zero. The only way that the product of these two brackets can be zero is if either 𝑥 plus
two is equal to zero or 𝑥 minus five is equal to zero. We can then solve each equation.
We’ll solve the first equation by subtracting two from both sides. That gives us that 𝑥 is equal to negative two. For the second equation, we’ll add five to both sides, which gives us 𝑥 is equal to
The next step is a little tricky. We need to sketch the curve of 𝑦 equals 𝑥 squared minus three 𝑥 minus 10. Remember by solving the equation when it was equal to zero, we found the roots of the
equation. That’s the point where the graph crosses the 𝑥-axis. A quadratic curve that has a positive coefficient for 𝑥 squared looks like a smiley
Finally, the 𝑦-intercept is given by the constant in our original equation — that’s
negative 10. We’re interested in the points on the graph where 𝑥 squared minus three 𝑥 minus 10
is less than or equal to zero — that’s the part of the curve that lies on and below
the 𝑥-axis. That occurs when 𝑥 is greater than or equal to negative two and less than or equal
to five. We can alternatively write that as 𝑥 is greater than or equal to negative two and
less than or equal to negative five in one inequality.