Inequality deals with a *range* of values, rather than an *exact number*, so it is necessary to replace the equal sign in an equation with its supposed inequality (either >, <, __>__, or __<)__.

For example, Solve for the following inequality:

**(x + 10) (2x – 9) > 0**

- Break up both of the brackets into two separate “sections”,
__x +12>__and__0__.__2x - 9 >0__ - So far, we know that each individual section must exceed the value of 0, as this is the inequality. From this, we can isolate the x value to determine the roots.

x+ 12 > 0__x > -10__

2x-9 >0

= 2x> 9__x > 9/2__

**(x + 10) (2x – 9) = 0**areand__x = -10, x = 9/2__,__x = 0__

__ __

- We can then break up number line into three intervals in relation to the roots that we have found.

Because the lowest root is -10, we recognize in order for the graph to continue negatively, x must therefore be less than -10, so x < -10.

Likewise, the highest root value is 9/2 so in order for the graph to continue positively, x must be bigger than 9/2, so 9/2 > x. We must also notice that the middle interval must (graphically) be larger than -10, yet smaller than 9/2, so, respectively, the inequality is represented by -10 <>

- You can test out each interval by plugging it into the equation and determining which one is the solution.

For x<-10, test__x = -11__

(-11 + 10) (2 (-11) -9) = 31

because the value “31” is greater than “0”, x< -10 is a solution. For -10<9/2,>x = 1

(1+10) (2(1)-9) = -77

because the value “-77” is less than “0”, -10 <>solution. For x > 9/2, test __x = 5__

(5+10) (2(5)-9) = 135

because the value “135” is greater than “0”, x>9/2 is a solution.

- Summarize the information found onto a table:

Interval Factor

x < -10

-10

x > 9/2

(x +10)

( - )

( + )

( + )

(2x – 9)

( - )

( - )

( + )

f(x)

**( + )**( - )

**( + )** - The solution, shown on the number line, is x < -10 and x > 9/2
**Note: Page 8 of our textbooks summarizes all possible intervals**

*Therefore, the solution is XE(-∞, -10] U[9/2, ∞)*

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