Find the first rational number between -2 and -3.

Mathematics
Find the first rational number between -2 and -3.

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Answer

−52,−94,−114-\frac{5}{2}, -\frac{9}{4}, -\frac{11}{4}

Step 1: Find the first rational number between −2-2 and −3-3. We use the formula for the midpoint of two numbers, 12(a+b)\frac{1}{2}(a+b). Let a=−2a = -2 and b=−3b = -3. x1=12(−2+(−3))x_1 = \frac{1}{2}(-2 + (-3)) x1=12(−2−3)x_1 = \frac{1}{2}(-2 - 3) x1=12(−5)x_1 = \frac{1}{2}(-5) x1=−52x_1 = -\frac{5}{2}

Step 2: Find the second rational number. Now, find a rational number between −2-2 and −52-\frac{5}{2}. Let a=−2a = -2 and b=−52b = -\frac{5}{2}. x2=12(−2+(−52))x_2 = \frac{1}{2}\left(-2 + \left(-\frac{5}{2}\right)\right) To add these, find a common denominator: x2=12(−42−52)x_2 = \frac{1}{2}\left(-\frac{4}{2} - \frac{5}{2}\right) x2=12(−92)x_2 = \frac{1}{2}\left(-\frac{9}{2}\right) x2=−94x_2 = -\frac{9}{4}

Step 3: Find the third rational number. Now, find a rational number between −52-\frac{5}{2} and −3-3. Let a=−52a = -\frac{5}{2} and b=−3b = -3. x3=12(−52+(−3))x_3 = \frac{1}{2}\left(-\frac{5}{2} + (-3)\right) To add these, find a common denominator: x3=12(−52−62)x_3 = \frac{1}{2}\left(-\frac{5}{2} - \frac{6}{2}\right) x3=12(−112)x_3 = \frac{1}{2}\left(-\frac{11}{2}\right) x3=−114x_3 = -\frac{11}{4}

The three rational numbers between −2-2 and −3-3 are −52,−94,−114\boxed{-\frac{5}{2}, -\frac{9}{4}, -\frac{11}{4}}.

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Find the first rational number between -2 and -3. We use the formula for the midpoint of two numbers, (1)/(2)(a+b).

Find the first rational number between -2 and -3.
Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI Answer
Step 1: Find the first rational number between -2 and -3. We use the formula for the midpoint of two numbers, (1)/(2)(a+b). Let a = -2 and b = -3. x_1 = (1)/(2)(-2 + (-3)) x_1 = (1)/(2)(-2 - 3) x_1 = (1)/(2)(-5) x_1 = -(5)/(2) Step 2: Find the second rational number. Now, find a rational number between -2 and -(5)/(2). Let a = -2 and b = -(5)/(2). x_2 = (1)/(2)(-2 + (-(5)/(2))) To add these, find a common denominator: x_2 = (1)/(2)(-(4)/(2) - (5)/(2)) x_2 = (1)/(2)(-(9)/(2)) x_2 = -(9)/(4) Step 3: Find the third rational number. Now, find a rational number between -(5)/(2) and -3. Let a = -(5)/(2) and b = -3. x_3 = (1)/(2)(-(5)/(2) + (-3)) To add these, find a common denominator: x_3 = (1)/(2)(-(5)/(2) - (6)/(2)) x_3 = (1)/(2)(-(11)/(2)) x_3 = -(11)/(4) The three rational numbers between -2 and -3 are -(5)/(2), -(9)/(4), -(11)/(4). That's 2 down. 3 left today — send the next one.