Question

which shows $-1\frac{3}{4}$ graphed on the number line? number lines with points at different positions

Explanation:

Step1: Understand the number

First, convert the mixed number \(-1\frac{3}{4}\) to an improper fraction or decimal. \(-1\frac{3}{4}=-\frac{7}{4}=-1.75\). This number is between \(-2\) and \(-1\) because \(-2 < -1.75 < -1\).

Step2: Analyze the number lines

- The first number line has a dot near \(-1\), which is greater than \(-1.75\) (closer to \(-1\) than \(-2\), but \(-1.75\) is closer to \(-2\) than \(-1\)? Wait, no: \(-1.75\) is 0.75 units away from \(-2\) (\(-2 + 0.75 = -1.25\)? Wait, no: distance from \(-2\) to \(-1.75\) is \(|-2 - (-1.75)| = 0.25\), and from \(-1.75\) to \(-1\) is \(|-1 - (-1.75)| = 0.75\). So it's closer to \(-2\) than to \(-1\). Wait, no: \(-1.75\) is between \(-2\) and \(-1\), 3/4 of the way from \(-2\) to \(-1\)? Wait, the interval between \(-2\) and \(-1\) is length 1. \(-1\frac{3}{4}\) is 3/4 units to the left of \(-1\), or 1/4 unit to the right of \(-2\) (since \(-2 + 1/4 = -1.75\)). So on the number line, between \(-2\) and \(-1\), the point should be 1/4 away from \(-2\) (towards \(-1\)).

Looking at the options:

- First graph: dot near \(-1\) (between \(-1\) and 0? No, between \(-2\) and \(-1\), closer to \(-1\)).

- Second graph: dot between \(-2\) and \(-1\), closer to \(-2\) (since \(-1.75\) is 0.25 from \(-2\) and 0.75 from \(-1\), so the dot here is between \(-2\) and \(-1\), which matches \(-1\frac{3}{4}\)).

- Third graph: dot between \(-1\) and 0? No, between \(-1\) and 0, which is positive side of \(-1\), but \(-1\frac{3}{4}\) is negative, left of \(-1\).

- Fourth graph: dot near \(-3\), which is wrong.

Wait, maybe I messed up. Let's re-express \(-1\frac{3}{4}\): it's \(-1 - \frac{3}{4}\), so on the number line, start at \(-1\), move 3/4 of a unit to the left (towards negative direction). The distance between \(-2\) and \(-1\) is 1 unit, divided into 4 parts (since denominator is 4), each part is 1/4. So from \(-2\) to \(-1\), the marks would be \(-2, -1\frac{3}{4}, -1\frac{2}{4}, -1\frac{1}{4}, -1\). Wait, no: between \(-2\) and \(-1\), the fraction \(\frac{3}{4}\) from \(-2\) towards \(-1\) is \(-2 + \frac{3}{4}(-1 - (-2)) = -2 + \frac{3}{4}(1) = -1\frac{1}{4}\)? No, wait, no: to find the position of \(-1\frac{3}{4}\) between \(-2\) and \(-1\): the value is \(-1.75\), so as a decimal, between \(-2\) (-2.0) and \(-1\) (-1.0). So \(-1.75\) is 0.75 below \(-1\), or 0.25 above \(-2\) (since \(-2 + 0.25 = -1.75\)). So on the number line, between \(-2\) and \(-1\), 1/4 of the way from \(-2\) to \(-1\) (since 0.25 is 1/4 of 1 unit). So the second graph has the dot between \(-2\) and \(-1\), which is where \(-1\frac{3}{4}\) should be. Wait, but let's check again:

Wait, \(-1\frac{3}{4}\) is equal to \(-1.75\). So on the number line, the integers are at -5, -4, -3, -2, -1, 0, etc. The distance between -2 and -1 is 1 unit. So -1.75 is 0.75 units to the left of -1, or 0.25 units to the right of -2. So on the number line, between -2 and -1, closer to -2. So the second graph (the one with the dot between -2 and -1) is the correct one. Wait, but the first graph's dot is between -2 and -1? No, first graph: the dot is between -2 and -1? Wait, first graph: the ticks are at -5, -4, -3, -2, -1, 0, etc. The dot is between -2 and -1? Wait, no: first graph's dot is at -1.5? No, first graph: the dot is between -2 and -1? Wait, the first graph: the tick marks are at -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. The dot is between -2 and -1? Wait, no, the first graph's dot is at -1.5? No, looking at the first graph: the dot is closer to -1, between -2 and -1 but closer to -1. The second graph's dot is between -2 and -1, closer to -2. Since -1.75 is 0.25 from -2 and 0.75 from -1, it's closer to -2, so the second graph (the middle one, second in the list) is correct. Wait, but the user's image: let's parse the graphs:

First graph: dots between -2 and -1, closer to -1.

Second graph: dots between -2 and -1, closer to -2.

Third graph: dots between -1 and 0.

Fourth graph: dots near -3.

So \(-1\frac{3}{4} = -1.75\), which is between -2 and -1, 0.25 away from -2 (so 1/4 of the interval from -2 to -1). So the second graph (the one with the dot between -2 and -1, closer to -2) is the correct one. Wait, but maybe I made a mistake. Let's confirm: \(-1\frac{3}{4}\) is less than -1 (since it's -1 minus 3/4), so it's to the left of -1, between -2 and -1. So the correct graph is the second one (the middle graph, the second in the list of four).

Answer:

The second number line (the one with the dot between \(-2\) and \(-1\))