Solve five 𝑥 squared minus three
𝑥 minus two is less than or equal to zero.
In order to be able to solve this
inequality, we first need to solve five 𝑥 squared minus three 𝑥 minus two is equal
to zero. To do this, we start by factorising
fully. Since both five and two are prime
numbers, we don’t have a huge number of options for the numbers inside our
First, we know that, at the front
of each bracket, we must have a five 𝑥 and an 𝑥, since five 𝑥 multiplied by 𝑥
gives us five 𝑥 squared. We also know that the only factors
of two are two and one.
To decide which order we choose, we
can use a little trial and error. Five 𝑥 multiplied by one is five
𝑥. And two multiplied by 𝑥 is two
𝑥. Two 𝑥 minus five 𝑥 gives us the
negative three we need and also ensures we get negative two rather than positive two
as our constant.
Remember, for the product of this
pair of brackets to be zero, either five 𝑥 plus two is equal to zero or 𝑥 minus
one is equal to zero. For the first value of 𝑥, let’s
start by subtracting two from both sides, then dividing by five. 𝑥 is equal to negative
two-fifths. For the second value of 𝑥, we can
just add one to both sides. 𝑥 is equal to one.
This last step is a little bit
tricky. We sketch the curve of 𝑦 equals
five 𝑥 squared minus three 𝑥 minus two. Remember, the 𝑥 values we worked
out are the roots of the equation. They’re the points where the graph
crosses the 𝑥-axis.
A quadratic curve with a positive
coefficient of 𝑥 squared is a smiley face. The 𝑦-intercept is given by the
constant negative two. Our inequality is five 𝑥 squared
minus three 𝑥 minus two is less than or equal to zero. This is the part of the curve that
lies on and below the 𝑥-axis. This occurs when 𝑥 is greater than
or equal to negative two-fifths and when 𝑥 is less than or equal to one.
The values of 𝑥 that satisfy our
inequality are, therefore, 𝑥 is greater than or equal to negative two-fifths and
less than or equal to one.