according to the triangle inequality, the length of the longest side of a triangle bust be less than the sum of the lengths of the other two sides. the two shortest sides of a triangle measure 3/10 and x + 1/5 and the longest measures 3/2x. write an inequality that uses the triangle inequality to relate the three sides
3x/2>3/10+x+1/5
Hmm. I get
3x/2 < 3/10+x+1/5
To use the triangle inequality, we need to consider the lengths of all three sides of the triangle.
Let's denote the two shortest sides as a and b, and the longest side as c.
Given that the two shortest sides measure 3/10 and x + 1/5, we can assign them as follows:
a = 3/10
b = x + 1/5
The longest side measures 3/2x, so we assign it as:
c = 3/2x
According to the triangle inequality, the sum of the lengths of the two shortest sides (a and b) must be greater than the length of the longest side (c). Therefore, we can write the triangle inequality as:
a + b > c
Substituting the assigned values, we have:
3/10 + x + 1/5 > 3/2x
To simplify this inequality, we need to find a common denominator for the fractions. The least common multiple (LCM) of 10 and 5 is 10, so we can rewrite the inequality as:
(3/10) + (2/10)(x + 1) > (3/2)x
Combining the fractions, we get:
(3 + 2(x + 1))/10 > (3/2)x
Expanding the numerator, we have:
(3 + 2x + 2)/10 > (3/2)x
Simplifying further, we get:
(2x + 5)/10 > (3/2)x
To get rid of the fraction, we can multiply both sides by the denominator (10):
10(2x + 5)/10 > 10(3/2)x
This simplifies to:
2x + 5 > 15x/2
Multiplying both sides by 2 to get rid of the fraction gives:
4x + 10 > 15x
Rearranging the terms, we have:
15x - 4x < 10
Combining like terms, we get:
11x < 10
To isolate x, we divide both sides by 11:
x < 10/11
Therefore, the inequality that relates the three sides of the triangle according to the triangle inequality is:
x < 10/11