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Chapter 3: Problem 59

Solve the inequality and sketch the solution set on a number line. $$-1 \leq y-3 \leq 2$$

Short Answer

Expert verified
The solution is \[\begin{equation} 2 \leq y \leq 5 onumber \end{equation}\].Sketch a number line from 2 to 5 with shaded region in between.

Step by step solution

01

- Isolate the variable

To solve the inequality \[\begin{equation} -1 \leq y-3 \leq 2 onumber \end{equation}\], first isolate the variable by adding 3 to all parts of the inequality. This gives\[\begin{equation} -1 + 3 \leq y - 3 + 3 \leq 2 + 3 onumber \end{equation}\].
02

- Simplify the inequality

Simplify the expressions in the inequality: \[\begin{equation} 2 \leq y \leq 5 onumber \end{equation}\].
03

- Sketch the solution set

To sketch the solution set, draw a number line. Mark the numbers 2 and 5 on the number line with closed circles because the inequality includes the equals sign. Shade the region between 2 and 5 on the number line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inequality solution
Solving an inequality involves finding the range of values that satisfy the condition given. When we solve \(-1 \leq y-3 \leq 2\), we're determining all possible values of \(y\) that make the inequality true. Unlike equations, where we find specific values that balance both sides, inequalities often give us a range of values.
In this problem, the inequality shows that the outcome should lie between -1 and 2 when subtracted by 3.
It's crucial to manipulate the inequality correctly to find the right solution range.
number line representation
A number line representation is a visual way to show the solution set of an inequality.
For the solved inequality \(2 \leq y \leq 5\), we need to sketch this on a number line. This helps us and others see the range of solutions quickly.
Here's how you do it:
  • Draw a horizontal line.
  • Mark the numbers 2 and 5 on the line.
  • Use closed circles on 2 and 5 since \(y\) can equal these values (denoted by \leq or \geq).
  • Shade the region between 2 and 5 to show all numbers that satisfy the inequality.

This shaded area represents the complete solution set visually.
algebraic manipulation
Algebraic manipulation involves using algebraic techniques to solve expressions or inequalities. To isolate the variable, we need to adjust the inequality step-by-step. Here鈥檚 how:
  • Start from the inequality \( -1 \leq y-3 \leq 2 \).
  • Add 3 to each part of the inequality to eliminate the -3 next to y.
  • This becomes \( -1+3 \leq y-3+3 \leq 2+3 \).
  • Simplify each part: \(2 \leq y \leq 5\).

Through these steps, you can see that algebraic manipulation simplifies and solves the inequality logically.

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