Answer:
[tex]-1\leq y-3 \:\:\textsf{ and }\:\: y-3 \leq 1[/tex]
Step-by-step explanation:
The absolute value of a number is its positive numerical value.
- To solve an equation containing an absolute value:
- Isolate the absolute value on one side of the equation.
- Set the contents of the absolute value equal to both the positive and negative value of the number on the other side of the equation.
- Solve both equations.
Given inequality:
[tex]|y-3|-4\leq -3[/tex]
Isolate the absolute value on the left side by adding 4 to both sides of the equation:
[tex]\implies |y-3|-4+4\leq -3+4[/tex]
[tex]\implies |y-3|\leq 1[/tex]
Case 1: (y - 3) is negative
[tex]\begin{aligned} \implies -(y-3) & \leq 1 \\ -y+3 & \leq 1\end{aligned}[/tex]
Multiply both sides by -1 (remembering to change the direction of the sign):
[tex]\implies y-3 & \geq -1[/tex]
Rearrange:
[tex]\implies -1\leq y-3[/tex]
Case 2: (y - 3) is positive
[tex]\implies y-3 & \leq 1[/tex]
Therefore, the compound inequality that represents the given inequality is:
[tex]-1\leq y-3 \:\:\textsf{ and }\:\: y-3 \leq 1[/tex]
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