Question 6470: Hi there,
I am confronted with the following question:
"Solve the inequality and express the solution in terms of intervals: (4x-3)(x+7)<=0."
Now when I go to solve this, I would do the following:
4x<=3
x<=3/4
And step 2:
x<=-7
Why is that wrong?
In solving inequalities, I thought the rule of thumb is to only flip the inequality sign when you multiply or divide by a negative. However, in order to get the correct answer for this question I would need to switch the inequality sign in step 2, even though all I am doing is subtracting 7 on both sides.
Thanks in advance,
Mike
Answer by longjonsilver(2297)   (Show Source):
You can put this solution on YOUR website! in learning maths, you have to understand what you are doing..not just the mechanics of working with inequalities, which you are OK with , but the understanding of what (4x-3)(x+7)<=0 is.
This is a quadratic equation, as such it is a u-shape. This u-shape, when plotted, crosses the x-axis at 2 points called the roots/solutions.
For a u-shape:
the curve is GREATER than zero, to the left of the lefthand root and also to the right of the righthand root.
the curve is LESS than zero, between the 2 roots.
So, understading this, you need to find your roots first and then "picture them" on a graph and the answer will be the ragion between them. What you have done is ASSUME something about your roots: you have assumed that both brackets are <=0..it does not say that, it says that BOTH BRACKETS MULTIPLIED TOGETHER is <=0 :-)
OK... to find the roots: --> (4x-3)(x+7)=0
so, either 4x-3=0 OR x+7=0
so x = 3/4 OR x=-7
so, where is the curve negative Mike? Visualise the curve: it is negative between -7 and 3/4. That is your answer.
jon.
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