document.write( "Question 175945: BUILDING A SKI RAMP. WRITE AN EQUALITY IN THE VARIABLE X FOR THE DEGREE MEASURE OF THE SMALLEST ANGLE OF THE TRIANGLE, GIVEN THE DEGREE MEASURE OF THE SMALLEST ANGLE IS AT MOST 30 DEGREE. First ANGLE =X, Second ANGLE = X+8 \n" ); document.write( "
Algebra.Com's Answer #131061 by gonzo(654)\"\" \"About 
You can put this solution on YOUR website!
let x be the smallest angle.
\n" ); document.write( "let x + 8 be the second angle.
\n" ); document.write( "let x + y = the third angle.
\n" ); document.write( "-----
\n" ); document.write( "\"x+%2B+x+%2B+8+%2B+y+=+180\" since the sum of the angles of a triangle equal 180 degrees.
\n" ); document.write( "this simplifies to \"2%2Ax+%2B+8+%2B+y+=+180\"
\n" ); document.write( "this simplifies to \"x+=+%28180-y-8%29%2F2\"
\n" ); document.write( "where:
\n" ); document.write( "x>0
\n" ); document.write( "x<=30
\n" ); document.write( "---
\n" ); document.write( "if x >0, then:
\n" ); document.write( "(180-y-8)/2 > 0
\n" ); document.write( "multiply both sides of this equation by 2 to get:
\n" ); document.write( "180 - y - 8 > 0
\n" ); document.write( "add y to both sides of the equation and combine like terms to get:
\n" ); document.write( "172 > y
\n" ); document.write( "subtract y from both sides of this equation and subtract 172 from both sides of this equation to get:
\n" ); document.write( "-y > - 172
\n" ); document.write( "multiply both sides of this equation by -1 to get:
\n" ); document.write( "y < 172
\n" ); document.write( "please note that multiplying both sides of an inequality by a minus number changes the inequality.
\n" ); document.write( "---
\n" ); document.write( "so we know that y has to be smaller than 172.
\n" ); document.write( "---
\n" ); document.write( "if x <= 30, then:
\n" ); document.write( "(180-y-8)/2 <= 30
\n" ); document.write( "multiply both sides by 2 to get:
\n" ); document.write( "180 - y - 8 <= 60
\n" ); document.write( "add y to both sides and subtract 60 from both sides to get:
\n" ); document.write( "180 - 60 - 8 <= y
\n" ); document.write( "combine like terms to get:
\n" ); document.write( "112 <= y
\n" ); document.write( "subtract 112 from both sides of the equation and subtract y from both sides of the equation to get:
\n" ); document.write( "-y <= -112
\n" ); document.write( "multiply both sides of the equation by -1 to get:
\n" ); document.write( "y >= 112
\n" ); document.write( "---
\n" ); document.write( "looks like x will be in the range of:
\n" ); document.write( "0 < x <= 30
\n" ); document.write( "when y is in the range of:
\n" ); document.write( "112 <= y < 172
\n" ); document.write( "---\r
\n" ); document.write( "\n" ); document.write( "to prove this is true, take several values of y in and out of the acceptable range and see what happens to x.
\n" ); document.write( "---
\n" ); document.write( "when y = 172, x = 0.
\n" ); document.write( "since y is outside of the acceptable range, x is also outside of the acceptable range so this is good.
\n" ); document.write( "---
\n" ); document.write( "when y = 112, x = 30.
\n" ); document.write( "since y is within the acceptable range, x is also within the acceptable range so this is good.
\n" ); document.write( "---
\n" ); document.write( "when y = 150, x = 11.
\n" ); document.write( "since y is within the acceptable range, x is also within the acceptable range so this is good.
\n" ); document.write( "---
\n" ); document.write( "when y = 200, x = -14.
\n" ); document.write( "since y is outside of the acceptable range, x is also outside of the acceptable range so this is good.
\n" ); document.write( "---
\n" ); document.write( "when y = 110, x = 31.
\n" ); document.write( "since y is outside of the acceptable range, x is also outside of the acceptable range so this is good.
\n" ); document.write( "---
\n" ); document.write( "the limits for y lead to the proper limits for x so the answer is good.
\n" ); document.write( "the equation to solve this problem is:
\n" ); document.write( "\"x+=+%28180-y-8%29%2F2\"
\n" ); document.write( "the requirements for x are:
\n" ); document.write( "0 < x <= 30
\n" ); document.write( "the requirements for y in order to satisfy the requirements for x are:
\n" ); document.write( "112 <= y < 172
\n" ); document.write( "---
\n" ); document.write( "as long as y is in the acceptable range, x will be in the acceptable range.
\n" ); document.write( "as long as y is outside of the acceptable range, x will be outside of the acceptable range.
\n" ); document.write( "---
\n" ); document.write( "since they asked for the equality equation for the smallest angle, your answer has to be:
\n" ); document.write( "\"x+=+%28180-y-8%29%2F2\"
\n" ); document.write( "with the stipulation that:
\n" ); document.write( "112 <= y < 172
\n" ); document.write( "---
\n" ); document.write( "i don't think you can solve this problem without introducing another variable since the range of that variable is important to the solution.
\n" ); document.write( "---
\n" ); document.write( "if you had stipulated what the third angle had to be in terms of x, that would have been different.
\n" ); document.write( "then we would have had a value for y and we could have used that value in place of y.
\n" ); document.write( "---\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
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