Question

solve for y and graph the solution.
1 > |y - 7|
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.
(there is a number line with marks 0,1,2,3,4,5,6,7,8,9,10,11,12 below the text)

Explanation:

Step1: Recall absolute value inequality rule

For \( |a| < b \) (where \( b>0 \)), it is equivalent to \( -b < a < b \). Here, the inequality is \( 1 > |y - 7| \), which can be rewritten as \( |y - 7| < 1 \). So we apply the rule with \( a = y - 7 \) and \( b = 1 \).

Step2: Solve the compound inequality

Using the rule, we get \( -1 < y - 7 < 1 \). Now, add 7 to all parts of the compound inequality.
For the left part: \( -1 + 7 < y - 7 + 7 \), which simplifies to \( 6 < y \).
For the right part: \( y - 7 + 7 < 1 + 7 \), which simplifies to \( y < 8 \).
Combining these, we have \( 6 < y < 8 \).

Answer:

The solution for \( y \) is \( 6 < y < 8 \). To graph this, we use an open circle at 6 and an open circle at 8, then draw a line segment between them (since \( y \) is strictly between 6 and 8, not including the endpoints).