In Between
Can you find the solution to this algebraic inequality?
Age
16 to 18
Challenge level
Problem
Find the range of values of $x$ for which
$$
{\sqrt{x}+ {{1}\over{\sqrt{x}}}} < {4}\,,
$$
where $\sqrt{x}$ is the positive root.
Getting Started
Rearrange the inequality to give a quadratic inequality.
Student Solutions
Well done to Minhaj, Amrit, Adithya, Sasi, Will, Charlie, Luke and Luke, Eric, Pablo, Julian, Johnny, Peter, Gabriel, and Manolis for their hard work on this problem!
Peter made the following observation:
I worked out that the range of numbers is one to three if you are talking about positive intergers.
Gabriel used a graphical method to solve the inequality.
Others used variations of the following method:
Peter made the following observation:
I worked out that the range of numbers is one to three if you are talking about positive intergers.
Gabriel used a graphical method to solve the inequality.
Others used variations of the following method:
Square both sides. Multiply throughout by x. Rearrange to form the quadratic inequality:
$$x^2-14x+1< 0.$$
Use the quadratic formula to solve this inequality. From the graph of $y =x^2 -14x + 1$ we see that the solution is $7-4\sqrt 3 < x < 7+4\sqrt 3$ or approximately $0.072< x< 13.928.$
Click on the pdf links to read Eric's, Minhaj's, Manolis's and Sasi's solutions.
Teachers' Resources
Why do this problem?
This is a non standard example of a quadratic inequality, the solution of which will involve algebraic manipulation. It can be used to help learners to practise the skills of estimation and approximation prior to engaging in an algebraic solution.Possible approach
There are various ways in which this problem can be tackled.
It can obviously be set 'straight' to learners, or used in a
slightly wider context, as described here.
To begin with, can learners assess a rough range of values
that is likely to emerge upon solving the inequality? For example,
would very large or very small values of $x$ satisfy the
inequality? As a class, who can spot the largest or smallest values
which will satisfy the inequality. This might naturally lead to a
numerical investigation, although in this case it turns out that
there is an exact algebraic solution to the problem.
Once the problem has some sense of a numerical estimation,
encourage the class to move on to an algebraic solution. Learners
will soon discover that they need to deal with the awkward
$\sqrt{x}$ terms. One way is to square both sides of the
inequality; another is to make the substitution $\sqrt{x} = p$. The
inequality can then be rearranged into a quadratic
inequality.
It is important for learners not just to blindly apply rules
of algebraic manipulation to the resulting inequalities. For each
manipulation (squaring, rearranging etc.) learners should explain
clearly why the operation preserves the inequality sign.
The next step will be to try to solve the quadratic
inequality. Learners might use an algebraic approach or a graphical
approach, but in each case will be required to use the quadratic
equation formula to find the solutions. It may be helpful for the
class to work together to solve the inequality. A fruitful class
discussion, where the learners are able to volunteer suggestions
for the steps in the working, is more likely to ensue if they have
at least started solving the problem for themselves and have worked
through the earlier steps.
Finally, once a solution is found it is good practice to check that
the boundaries of the inequality work and also to compare these to
the original estimates.Key questions
Can you estimate an approximate range of values for which the
inequality is satisfied?
What things might we try to remove the awkward $\sqrt{x}$
term?
How do we find the factors of a quadratic expression?
If you know that the product of two factors is negative what can you say about the factors?