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Solving Quadratics with Inequalities

Q:  Solve for x:  (x + 3)(x – 4) < 0

A:

Step 1:  Find the zeroes

Since the quadratic is already factored, this isn’t too tricky.  If it wasn’t factored, you’d have to factor first! (always make sure there is a 0 on one side of the equation / inequality before proceeding).

OK, so what are the zeroes?

x + 3 = 0 or x – 4 = 0

The zeroes are x = -3 or x = 4

So, draw a number line and plot the zeroes on the number line:

numberline

Now, you have to test each “interval” that is separated by the “zeroes”.  There are three intervals to test.

Interval 1:  The numbers to the left of -3  –> in interval notation this is (-infinity, -3)

Interval 2:  The numbers between -3 and 4  –> in interval notation this is (-3, 4)

Interval 3:  The numbers to the right of 4  –> in interval notation this is (4, infinity)

Step 2:  Test each interval

Pick any number on interval 1 and test it into the original inequality.  I’ll pick -5:

(x + 3)(x – 4) < 0

(-5 + 3)(-5 – 4) < 0

(-2)(-9) < 0

18 < 0      <—  this is false, so numbers on this interval [interval 1] are not part of the solution.

Pick any number on interval 2 and test it into the original inequality.  I’ll pick 0:

(x + 3)(x – 4) < 0

(0 + 3)(0 – 4) < 0

(3)(-4) < 0

-12 < 0      <—  this is true, so numbers on this interval [interval 2] are a part of the solution.

Pick any number on interval 3 and test it into the original inequality.  I’ll pick 5:

(x + 3)(x – 4) < 0

(5 + 3)(5 – 4) < 0

(8)(1) < 0

8 < 0      <—  this is false, so numbers on this interval [interval 3] are not part of the solution.

SO:  The only interval that “worked” was interval 2.  Therefore, the solution is all number between -3 and 4.

In interval notation, we write that like: (-3, 4)

In inequality notation, we write that like: -3 < x < 4

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Solving quadratics with inequalities

Q:  Solve  (a/2)2 – a > 0

A:  Follow the steps below

Solving Algebraically:

Step 1:  Simplify

We can simplify the equation by squaring the fraction to get:

a2/4 – a > 0

OR in different form:

(1/4) a2 – a > 0

Now, to get rid of the fraction and clean things up, I am going to multiply everything by 4 (since the fraction is 1/4 — you don’t have to do this, just a “cleaning up” step”):

4*(1/4) a2 – 4*a > 4*0

a2 – 4a > 0

Normally, when you solve inequalities, you isolate the variable by moving things around to the left / right side.  When it is a quadratic, you don’t want to do that.  You want all of the numbers and variables on one side and zero on the other side.  We have that, so we are good to go.

Step 2:  Factor

Now, factor the left side:

a2 – 4a > 0

a (a – 4) > 0

Step 3:  Identify the zeroes

As we are used to doing with quadratics, we need to find what values you plug in to make “zeroes”

So, take each factor and set it equal to zero like so:

a = 0 and a – 4 = 0

Solve to get:

a = 0 and a = 4

The zeroes are:  0 and 4.

So, this parabola (quadratic) crosses the x-axis at 0 and 4.  Now, we want to find where the quadratic is greater than 0.

With analysis, we know the quadratic is concave up in shape (a smiley face).  On your paper, draw an x-y graph with a parabola that crosses the x-axis and 0 and 4 and is concave up.  You should get:

So, where is the parabola > 0 (above the x-axis?)

When a < 0 and when a > 4