document.write( "Question 481190: Can you please help me solve this I really need help I have tried many times but I get stuck and i need to give it in very soon. \r
\n" ); document.write( "\n" ); document.write( "-What is the solution to the following system of inequalities?
\n" ); document.write( "2x > -5 + 3y
\n" ); document.write( "2x − 5y < -6 \n" ); document.write( "

Algebra.Com's Answer #329520 by Theo(13342) $\"\"$ $\"About$

You can put this solution on YOUR website!
solve for y in both equations to get:
\n" ); document.write( "y < (2x+5)/3
\n" ); document.write( "y > (2x+6)/5
\n" ); document.write( "set these equations to equality to get:
\n" ); document.write( "y = (2x+5)/3
\n" ); document.write( "y = (2x+6)/5
\n" ); document.write( "since they are both equal to y, then they are both equal to each other and you get:
\n" ); document.write( "(2x+5)/3 = (2x+6)/5
\n" ); document.write( "cross multiply to get:
\n" ); document.write( "3*(2x+6) = 5*(2x+5)
\n" ); document.write( "multiply out to get:
\n" ); document.write( "6x + 18 = 10x + 25
\n" ); document.write( "subtract 6x from both sides of the equation and subtract 18 from both sides of the equation to get:
\n" ); document.write( "-7 = 4x
\n" ); document.write( "divide both sides of the equation by 4 to get:
\n" ); document.write( "x = -7/4
\n" ); document.write( "when x = -7/4, both equations are equal to each other.
\n" ); document.write( "look at what happens when x is smaller than -7/4 and when x is > -7/4
\n" ); document.write( "i chose -2 for smaller and -1 for greater.
\n" ); document.write( "my results from solving for these values of x is shown in the following table.
\n" ); document.write( "

\r\n" );
document.write( "     x         y1 = (2x+5)/3       y2 = (2x+6)/5\r\n" );
document.write( "    -7/4             .5                  .5\r\n" );
document.write( "    -1               1                   .8\r\n" );
document.write( "    -2               .33333333           .4\r\n" );
document.write( "

\n" ); document.write( "from this table you can see that:
\n" ); document.write( "when x is equal to -7/4, y1 = y2
\n" ); document.write( "when x is greater than -7/4, y1 is greater than y2.
\n" ); document.write( "when x is smaller than -7/4, y1 is smaller than y2.
\n" ); document.write( "we were looking to solve for:
\n" ); document.write( "y < (2x+5)/3
\n" ); document.write( "y > (2x+6)/5
\n" ); document.write( "since y1 = (2x+5)/3 and y2 = (2x+6)/5, then we are looking for a value of y that is smaller than y1 and greater than y2.
\n" ); document.write( "this is possible when y1 is greater than y2 which occurs when x is greater than -7/4.
\n" ); document.write( "the equation is satisfied when x > -7/4.
\n" ); document.write( "if we graph both the equations of:
\n" ); document.write( "y = (2x+5)/3
\n" ); document.write( "y = (2x+6)/5
\n" ); document.write( "we will see that the region of compatibility is when x > -7/4 as shown below.
\n" ); document.write( " $\"graph%28400%2C400%2C-3%2C1%2C-1%2C3%2C%282x%2B5%29%2F3%2C%282x%2B6%29%2F5%29\"$
\n" ); document.write( "the graph that crosses the y-axis closer to 1 is the equation of y = (2x+5)/3.
\n" ); document.write( "that becomes the lower line after the crossing point of x = -7/4.
\n" ); document.write( "you get:
\n" ); document.write( "the higher line is y = (2x+6)/5
\n" ); document.write( "the lower line is y = (2x+5)/3
\n" ); document.write( "any value of y between these lines will simultaneously satisfy the equations of:
\n" ); document.write( "y < (2x+6)/5
\n" ); document.write( "y > (2x+5)/3
\n" ); document.write( "this only occurs when x is greater than -7/4.
\n" ); document.write( "since that's what we originally set out to find, then the solution is good.
\n" ); document.write( "our answer is that the value of x that satisfy both requirements occurs when x > -7/4.
\n" ); document.write( "it helps to find the crossover point first.
\n" ); document.write( "once you do that, you can test for values of x above and below that value.
\n" ); document.write( "since these are linear equations, you will only find one crossover point.
\n" ); document.write( "when you get to higher order equations, the crossover points can be more than 1.
\n" ); document.write( "you will then get regions where the requirements of the simultaneous equations are satisfied.
\n" ); document.write( "for this set of equations, there is only one region that satisfies the requirements and that is when the value of x is greater than -7/4.\r
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