To solve the absolute value inequality [tex]\( |2y - 3| \leq 0.4 \)[/tex], follow these steps:
1. Understand the Absolute Value Inequality:
The absolute value inequality [tex]\( |A| \leq B \)[/tex] can be rewritten as a double inequality:
[tex]\[
-B \leq A \leq B
\][/tex]
In this case, [tex]\( A = 2y - 3 \)[/tex] and [tex]\( B = 0.4 \)[/tex]. So we can rewrite the inequality:
[tex]\[
-0.4 \leq 2y - 3 \leq 0.4
\][/tex]
2. Separate the Double Inequality:
Now, split the double inequality into two separate inequalities:
[tex]\[
-0.4 \leq 2y - 3
\][/tex]
and
[tex]\[
2y - 3 \leq 0.4
\][/tex]
3. Solve the First Inequality:
[tex]\[
-0.4 \leq 2y - 3
\][/tex]
Add 3 to both sides of the inequality:
[tex]\[
-0.4 + 3 \leq 2y
\][/tex]
Simplify the left side:
[tex]\[
2.6 \leq 2y
\][/tex]
Divide both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[
1.3 \leq y
\][/tex]
4. Solve the Second Inequality:
[tex]\[
2y - 3 \leq 0.4
\][/tex]
Add 3 to both sides of the inequality:
[tex]\[
2y \leq 0.4 + 3
\][/tex]
Simplify the right side:
[tex]\[
2y \leq 3.4
\][/tex]
Divide both sides by 2 to solve for [tex]\( y \)[/tex]:
[tex]\[
y \leq 1.7
\][/tex]
5. Combine the Results:
Combine the solutions from the two inequalities to write the final solution:
[tex]\[
1.3 \leq y \leq 1.7
\][/tex]
Thus, the solution to the inequality [tex]\( |2y - 3| \leq 0.4 \)[/tex] is [tex]\( 1.3 \leq y \leq 1.7 \)[/tex].
