Solve the inequality 𝑥 plus nine times 𝑥 minus two is less than or equal to 22𝑥 minus 74. We’re going to solve this inequality in three steps.
Firstly, we’re going to rearrange the inequality that we have, putting it in the form 𝑓 of 𝑥 is less than or equal to zero. Then we’re going to sketch a graph of 𝑓 of 𝑥. And we’ll see the once we’ve graphed 𝑓 of 𝑥, we will be able to conclude.
So naturally we start with our first step. We write down the inequality that we have. We expand the brackets on the left-hand side, so 𝑥 plus nine times 𝑥 minus two becomes 𝑥 squared plus seven 𝑥 minus 18. We add 74 to both sides, so we get 𝑥 squared plus seven 𝑥 plus 56 on the left-hand side and just 22𝑥 on the right-hand side.
And finally we subtract 22𝑥 from both sides to get 𝑥 squared minus 15𝑥 plus 56 is less than or equal to zero. We have written the inequality in the required form for step one, so we’re now ready to move on to step two.
Okay, so now we have to graph 𝑓 of 𝑥. What is 𝑓 of 𝑥? Well it’s the left-hand side of the inequality that we got after completing step one. We can see that 𝑓 of 𝑥 is a quadratic function. And of course to graph a quadratic function, it’s very helpful to know the factors because they tell you the 𝑥- intercept of the graph.
So let’s attempt to factor 𝑓 of 𝑥. We’re looking for two numbers whose sum is negative 15 and whose product is 56. Two numbers which satisfy those requirements are negative seven and negative eight.
And so 𝑓 of 𝑥 when factored is 𝑥 minus seven times 𝑥 minus eight. So let’s try to sketch a graph. We can see that 𝑓 of 𝑥 has zeros at 𝑥 equals seven and 𝑥 equals eight, so we know its graph must pass through the points seven, zero and eight, zero, which are marked.
We also know that as 𝑓 of 𝑥 is a quadratic function, its graph will be 𝑎 parabola. So the only question is whether it’s an upward- or downward-facing parabola. How can we tell? Well for the graph of 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏 𝑥 plus 𝑐, if the coefficient of 𝑥 squared, 𝑎, is greater than zero, then we have an upward-facing parabola; and if 𝑎 is less than zero, then we have a down-facing parabola. So which case do we have here?
The coefficient of 𝑥 squared is just one, which is greater than zero, and so we have an upward-facing parabola. As a result, the graph of 𝑓 of 𝑥 looks something like this. Now we have a sketch of the graph of 𝑓 of 𝑥. Of course this sketch isn’t particularly accurate, but it’s good enough for what we need it for.
We can now conclude. The inequality we’re trying to solve now is 𝑓 of 𝑥 is less than or equal to zero. 𝑓 of 𝑥 is less than or equals to zero when its graph is below the 𝑥-axis, which happens between 𝑥 equals seven and 𝑥 equals eight.
So that’s the solution to the inequality informally, but we need to make it mathematical. First of all, we need to clarify what we mean by between, our 𝑥 equals seven and 𝑥 equals eight included in the solution set of this inequality.
Looking at the graph we can remember that 𝑓 of seven is zero and so is definitely less than or equal to zero; similarly, 𝑓 of eight is zero and so it satisfies 𝑓 of 𝑥 is less than or equal to zero. So 𝑥 equal seven and 𝑥 equals eight are included in the solution sets.
Here’s one way to express this mathematically; we say that seven is less than or equal to 𝑥 which is less than or equal to eight. And the fact that we are using less than or equal to signs instead of just less than signs tells us that seven and eight are allowed values of 𝑥.
We can also express this fact using set notation and interval notation. So we say that 𝑥 is in the interval from seven to eight. And the fact that we’re using square brackets here instead of round parentheses tells us that the endpoints seven and eight are included in this interval.
Let’s just recap what we’ve done. We took this inequality and rearranged it to the form 𝑓 of 𝑥 is less than or equal to zero; that was the first step. Had the inequality sign been just a less than sign, of course we would be rearranging to just 𝑓 of 𝑥 is less than zero. The important thing is that on the right-hand side of the inequality sign whatever it maybe, we have zero.
We then took this 𝑓 of 𝑥 and we sketched its graph, and we factored 𝑓 of 𝑥 to allow us to do this. Finally, we concluded from the graph what the solution was.
There were two things that we needed from the graph in order to be able to conclude: we needed the 𝑥-intercepts, seven and eight, and we also needed to know the orientation of the parabola which passes through these two points.
Had we drawn a downward-facing parabola, we’ve have got the wrong answer. To check that we’ve got the answer right, you might like to check that values inside the interval that we’ve got do indeed satisfy the original inequality that we had: 𝑥 plus nine times 𝑥 minus two is less than or equal to 22𝑥 minus 74. And further, you might like to check that values outside this interval do not satisfy the inequality.