SOLUTION: Solve the inequality: {{{6x^2+5x<4}}}
Here's what i did: {{{6x^2+5x<4}}}
{{{6x^2+5x-4<0}}}
{{{(2x-1)(3x+4)<0}}} Now what?
Algebra ->
Inequalities
-> SOLUTION: Solve the inequality: {{{6x^2+5x<4}}}
Here's what i did: {{{6x^2+5x<4}}}
{{{6x^2+5x-4<0}}}
{{{(2x-1)(3x+4)<0}}} Now what?
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You can put this solution on YOUR website! Solve the inequality:
Here's what i did: <===ok ... this is given <=== Step 1 ok <=== Step 2 ok
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What you've done so far is very good. You just need a little prompting to hopefully see
the light at the end of the tunnel.
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Let's back off Step 1 and Step 2 above and think about them more in the form of an equation.
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In a general sense we can say that a quadratic equation is of the form:
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From what you probably know about quadratic equations, you can tell that their graphs are
parabolas. If "a" (the multiplier of ) is positive the graph is loosely shaped like
the letter "U" ... as you look horizontally from left to right the graph descends to
a low point, and then turns and ascends towards infinity.
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Now think of your Step 1 as being in the form of a quadratic equation:
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Because the multiplier of the term is positive, the parabolic graph of this equation
will be U-shaped. If the graph crosses the x-axis, the coordinates of the points where it
crosses will have their y values equal to zero. So by setting y equal to zero we can find
the values of x that correspond. So set y equal to zero and solve the equation:
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You can now factor this equation just as you did in Step 2 to get:
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Then, with this quadratic polynomial equal to zero the equation will be true if either of
the two factors is equal to zero because a multiplication by zero on the left side makes
the entire left side equal to the zero on the right side.
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Therefore, if either or then the quadratic polynomial
will equal zero and the values of x that make that happen will be the values on the x-axis
where the graph intersects that axis. Solve these two equations and you get:
. <=== Add 1 to both sides and then divide both sides by 2 to get:
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and <=== Subtract 4 from both sides and then divide both sides by 3 to get:
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This tells us that the graph intersects the x-axis at and at
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For values of x between those two points the graph is sagging below the x-axis. (For values of
x greater than +1/2 or less than -4/3 the graph will be above the x-axis.)
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Now recall that in Step 1 you had the problem down to:
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You now know that as indicated by its graph, the polynomial on the left side will have
a y value of less than zero when the graph is in the "sagging" portion between the two
values of and . Any value of x between those two values will
make the polynomial have a value less than zero as indicated by the fact that the y
value (which equals the value of the polynomial and corresponds to the graph) will be
negative and therefore less than zero.
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We can write the answer to the problem as:
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and that's the answer you are looking for.
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Hope that something in this long discussion of the problem gets you thinking about a way
to visualize what this problem means in relation to the graph and how the graph can help
you to interpret the answer.
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