Question

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

Explanation:

Step1: Recall absolute value inequality rule

For \(|a| < b\) (where \(b>0\)), it is equivalent to \(-b < a < b\). Here, the inequality is \(3 > |y - 4|\), which can be rewritten as \(|y - 4| < 3\).

Step2: Apply the rule to the inequality

Using the rule, we get \(-3 < y - 4 < 3\).

Step3: Solve the left - hand side of the compound inequality

Add 4 to all parts of the inequality: \(-3+4 < y-4 + 4<3 + 4\).
For the left - hand side: \(-3 + 4=1\).
For the right - hand side: \(3+4 = 7\).
So, \(1 < y < 7\).

To graph this solution:

• We draw a number line.
• We place open circles at \(y = 1\) and \(y=7\) (because the inequality is strict, \(y\) cannot be equal to 1 or 7).
• Then we draw a line segment between the open circles at 1 and 7 to represent all the values of \(y\) that satisfy the inequality.

Answer:

\(1 < y < 7\)