document.write( "Question 474427: how can i answer this y > 1 + 3x
\n" ); document.write( " y < 2x + 4
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Algebra.Com's Answer #325381 by Theo(13342) $\"\"$ $\"About$

You can put this solution on YOUR website!
your problem is:
\n" ); document.write( "y > 3x + 1
\n" ); document.write( "y < 2x + 4
\n" ); document.write( "this can only happen if 2x + 4 > 3x + 1
\n" ); document.write( "that's the equation you have to solve.
\n" ); document.write( "subtract 1 from both sides of that equation to get:
\n" ); document.write( "2x + 3 > 3x
\n" ); document.write( "subtract 2x from both sides of that equation to get:
\n" ); document.write( "3 > x
\n" ); document.write( "this is the same as:
\n" ); document.write( "x < 3
\n" ); document.write( "to test this out, set up a table of values where the value of x is less than 3 and equal to 3 and greater than 3 and solve for each equation.
\n" ); document.write( "in this table, make:
\n" ); document.write( "y1 = 3x + 1
\n" ); document.write( "y2 = 2x + 4
\n" ); document.write( "since 2x + 4 needs to be greater than 3x + 1, this means that y2 needs to be greater than y1.
\n" ); document.write( "your table will look like this:
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document.write( "         x        y1        y2       remarks\r\n" );
document.write( "         0        1         4        y2 > y1 (ok)\r\n" );
document.write( "         2        7         8        y2 > y1 (ok)\r\n" );
document.write( "         3        10        10       y2 not > y1 (no good)\r\n" );
document.write( "         4        13        12       y2 not > y1 (no good)\r\n" );
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\n" ); document.write( "you can see from the table that the original statement is only true when y2 > y1 which only happens when x < 3.
\n" ); document.write( "you can graph the equations of y1 = 3x + 1 and y2 = 2x + 4 and the graph will also show you that you can only find a y between those 2 equations when the value of x is smaller than 3.
\n" ); document.write( "the graph is shown below:
\n" ); document.write( "the graph of y = 3x + 1 intersects the y-axis at y = 1.
\n" ); document.write( "the graph of y = 2x + 4 intersects the y-axis at y = 4.
\n" ); document.write( " $\"graph%28600%2C600%2C-5%2C5%2C-15%2C15%2C3x%2B1%2C2x%2B4%29\"$
\n" ); document.write( "a value of y < 2x + 4 and > 3x + 1 can only happen in the region where the line of 3x + 1 is lower than the line of 2x + 4.
\n" ); document.write( "that only happens when the value of x is smaller than 3.
\n" ); document.write( "you find that by picking any value of x < 3 and solving for each equation at that value of x.
\n" ); document.write( "example:
\n" ); document.write( "when x = -5, your equations of:
\n" ); document.write( "y > 3x + 1 and y < 2x + 4 become:
\n" ); document.write( "y > -14 and y < -6
\n" ); document.write( "you can see that, when x = -5, any value of y < -6 and > -14 can be found between the lines of the equations.
\n" ); document.write( "when x = 5, your equations of:
\n" ); document.write( "y > 3x + 1 and y < 2x + 4 become:
\n" ); document.write( "y > 16 and y < 14
\n" ); document.write( "you can see that this is an impossible situation, so, when x = 5, the system of equation is false.
\n" ); document.write( "the system of equations is true only if x < 3.
\n" ); document.write( "if you were able to, you would shade the region between the 2 lines for all values of x < 3.
\n" ); document.write( "that would be the region of compatibility with the system of equations.\r
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