a) Represent -5/3 < x < -1/2 on a number line.

Here are the number line representations for the given inequalities: a) Represent -5/3 < x < -1/2 on a number line. First, convert the fractions to decimals for easier placement: -5/3 ≈ -1.67 -1/2 = -0.5 So the inequality is approximately -1.67 < x < -0.5. To represent this on a number line: 1. Draw a horizontal line and label it as a number line. 2. Mark key integer points like -2, -1, and 0. 3. Locate -5/3 (between -2 and -1, closer to -1.5) and -1/2 (between -1 and 0). 4. Place an open circle at -5/3 because x is strictly greater than -5/3 (not equal to). 5. Place an open circle at -1/2 because x is strictly less than -1/2 (not equal to). 6. Shade the region between the two open circles to indicate all values of x that satisfy the inequality. The number line representation would look like this: ` <---------------------------------------------------------------------> -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. First, convert the fractions to decimals for easier placement: 1/2 = 0.5 7/2 = 3.5 So the inequality is 0.5 x < 3.5. To represent this on a number line: 1. Draw a horizontal line and label it as a number line. 2. Mark key integer points like 0, 1, 2, 3, and 4. 3. Locate 1/2 (between 0 and 1) and 7/2 (between 3 and 4). 4. Place a closed circle (or solid dot) at 1/2 because x is greater than or equal to 1/2. 5. Place an open circle at 7/2 because x is strictly less than 7/2 (not equal to). 6. Shade the region between the closed circle at 1/2 and the open circle at 7/2 to indicate all values of x that satisfy the inequality. The number line representation would look like this: ` <---------------------------------------------------------------------> 0 1/2 1 2 3 7/2 4 ●----------------------o ` (The line segment between the closed circle '●' and the open circle 'o' is shaded.)