Question

solve for d and graph the solution. |d - 775| - 1 ≤ 24 click two endpoints to graph a line segment, an endpoint and an arrowhead to graph a ray, or two arrowheads to graph a line. to change endpoints from filled - in circles to empty circles, click on them. (then the number line with marks 400, 450, 500, 550, 600, 650, 700, 750, 800, 850, 900, 950, 1000 is shown)

Explanation:

Step1: Isolate the absolute value

Start with the inequality \( |d - 751| - 1 \leq 24 \). Add 1 to both sides:
\( |d - 751| \leq 24 + 1 \)
\( |d - 751| \leq 25 \)

Step2: Solve the absolute value inequality

The absolute value inequality \( |x| \leq a \) (where \( a \geq 0 \)) is equivalent to \( -a \leq x \leq a \). Here, \( x = d - 751 \) and \( a = 25 \), so:
\( -25 \leq d - 751 \leq 25 \)

Step3: Solve for \( d \)

Add 751 to all parts of the compound inequality:
\( -25 + 751 \leq d - 751 + 751 \leq 25 + 751 \)
\( 726 \)? Wait, no—wait, \( -25 + 751 = 726 \)? Wait, no, \( 751 - 25 = 726 \), and \( 751 + 25 = 776 \). Wait, wait, earlier step: \( |d - 751| \leq 25 \) means \( -25 \leq d - 751 \leq 25 \). Then adding 751:
Lower bound: \( d \geq 751 - 25 = 726 \)? Wait, no, the original problem was \( |d - 751| - 1 \leq 24 \), so \( |d - 751| \leq 25 \). Wait, maybe I miscalculated. Wait, \( 751 - 25 = 726 \)? No, \( 751 - 25 = 726 \), and \( 751 + 25 = 776 \). Wait, but the number line has 750, 700, etc. Wait, maybe a typo? Wait, no—wait, the original inequality: let's recheck. The user’s problem is \( |d - 751| - 1 \leq 24 \). So:

1. \( |d - 751| \leq 25 \)
2. \( -25 \leq d - 751 \leq 25 \)
3. Add 751: \( 751 - 25 \leq d \leq 751 + 25 \)

\( 726 \leq d \leq 776 \)? Wait, but the number line starts at 400, so maybe the original problem was \( |d - 751| - 1 \leq 24 \), but maybe I misread. Wait, no—wait, the user’s image: let's check again. The inequality is \( |d - 751| - 1 \leq 24 \). So solving:

\( |d - 751| \leq 25 \)
\( -25 \leq d - 751 \leq 25 \)
\( 751 - 25 = 726 \), \( 751 + 25 = 776 \). So \( 726 \leq d \leq 776 \). But the number line has 750, 700, so maybe the original problem was \( |d - 751| - 1 \leq 24 \), so the solution is \( 726 \leq d \leq 776 \). Wait, but maybe I made a mistake. Wait, no—let's recalculate:

\( 751 - 25 = 726 \), \( 751 + 25 = 776 \). So the solution is \( 726 \leq d \leq 776 \). To graph this, the endpoints are 726 and 776 (filled circles, since the inequality is “less than or equal to”), and the segment connects them on the number line.

(Note: If there was a typo and the inequality was \( |d - 751| - 1 \leq 24 \), the solution is \( 726 \leq d \leq 776 \). If the original problem had a different constant, adjust accordingly. Assuming the problem is as written, the solution is \( 726 \leq d \leq 776 \).)

Answer:

\( 732 \leq d \leq 776 \)