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
Here are the number line representations for the given inequalities: a) Represent -5/3 < x < -1/2 on a number line. Step 1: Convert the fractions to decimals for easier placement. -(5)/(3) ≈ -1.67 -(1)/(2) = -0.5 The inequality is approximately -1.67 < x < -0.5. Step 2: Draw a number line and mark relevant points. Mark integer points such as -2, -1, and 0. Step 3: Place circles at the endpoints. Since the inequalities are strict (<), use open circles at -5/3 and -1/2. Step 4: Shade the region between the circles. The number line representation is: ` <---------------------------------------------------------------------> -2 -5/3 -1/2 0 1 o-------------o ` (The line segment between the two open circles 'o' is shaded.) b) Represent 1/2 x < 7/2 on a number line. Step 1: Convert the fractions to decimals for easier placement. (1)/(2) = 0.5 (7)/(2) = 3.5 The inequality is 0.5 x < 3.5. Step 2: Draw a number line and mark relevant points. Mark integer points such as 0, 1, 2, 3, and 4. Step 3: Place circles at the endpoints. Since x is greater than or equal to 1/2 (), use a closed circle (solid dot) at 1/2. Since x is strictly less than 7/2 (<), use an open circle at 7/2. Step 4: Shade the region between the circles. The number line representation is: ` <---------------------------------------------------------------------> 0 1/2 1 2 3 7/2 4 ●----------------------o ` (The line segment between the closed circle '●' and the open circle 'o' is shaded.)