Question 411433
{{{18x^2+9x-20 = 0}}}
To solve this equation we factor it (or use the Quadratic Formula). Factoring it we get:
(6x-5)(3x+4) = 0
From the Zero Product Property we know that one of these factors must be zero. So:
6x-5 = 0 or 4x+3 = 0
Solving each of these we get:
{{{x = 5/6}}} or {{{x = -4/3}}}<br>
To solve the inequalities we will use the answers to the equation. When {{{x = 5/6}}} the factor (6x-5) is zero. When x is greater than 5/6, the factor will be larger than zero. (IOW: positive). And when x is less than 5/6 the factor willbe smaller than zero (negative). Using the same logic on the other factor we find that when x is larger than -4/3 the (4x+3) factor will be positive and when x is smaller than -4/3, that factor will be negative.<br>
Now let's put this together. Think of a number line with the numbers -4/3 and 5/6 plotted. These two points divide the number line into three parts:<ul><li>The part to the left of -4/3</li><li>The part to the right of 5/6</li><li>The part between -4/3 and 5/6</li></ul>
Let's analyze what each factor, (6x-5) and (4x+3), will be in each of these parts of the number line:<ul><li>For x's to the left of -4/3, both factors will be negative. And since a negative times a negative is positive, (6x-5)(4x+3) will be positive.</li><li>For x's to the right of 5/6, both factors will be positive. And since a positive times a positive is positive, (6x-5)(4x+3) will be positive.</li><li>In between -4/3 and 5/6, (4x+3) will be positive and (6x-5) will be negative. And since a positive time a negative is negative, (6x-5)(4x+3) will be negative.</li></ul>
So we have found that if
{{{x < -4/3}}} or {{{x > 6/5}}}
then (6x-5)(4x+3) will be positive. If we include the values that make the factors zero we get:
{{{x <= -4/3}}} or {{{x >= 6/5}}}
This is the solution to {{{18x^2 + 9x - 20 >= 0}}}
In set notation this is {x | {{{x <= -4/3}}} or {{{x >= 6/5}}}}<br>
And if
{{{x > -4/3}}} and {{{x < 5/6}}}
then (6x-5)(4x+3) will be negative. If we include the values that make the factors zero we get:
{{{x >= -4/3}}} and {{{x <= 5/6}}}
This is the solution to {{{18x^2 + 9x - 20 <= 0}}}<br>
In set notation this is {x | {{{x >= -4/3}}} and {{{x <= 5/6}}}} or
{x | {{{-4/3 <= x <= 5/6}}}}