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Chapter 2: Problem 113
Graph the solution set, and write it using interval notation \(-12 \leq \frac{1}{2} z+1 \leq 4\)
Short Answer
Expert verified
-26 to 6, inclusive. Written as \([-26, 6]\)
Step by step solution
01
- Isolate the variable
Start by isolating the term with the variable in the given compound inequality ewline ewline \(-12 \leq \frac{1}{2}z + 1 \leq 4\)
02
- Subtract 1 from all parts
Subtract 1 from each part of the compound inequality to eliminate the constant on the right side of the middle term. ewline \(-12 - 1 \leq \frac{1}{2}z + 1 - 1 \leq 4 - 1\) ewline Simplifies to: ewline \(-13 \leq \frac{1}{2}z \leq 3\)
03
- Clear the fraction
Multiply all parts of the inequality by 2 to clear the fraction. ewline \(-13 \times 2 \leq \frac{1}{2}z \times 2 \leq 3 \times 2\) ewline Simplifies to: ewline \(-26 \leq z \leq 6\)
04
- Write the solution in interval notation
Write the solution set \(-26 \leq z \leq 6\) in interval notation as ewline \([-26, 6]\)
05
- Graph the solution set
Graph the solution set on a number line. For this interval, include solid dots at \(-26\) and \ \( -26 \) \ (( and z≤\b MathInequal:/]
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a way to write subsets of real numbers. It uses brackets \([ ]\) to include endpoints and parentheses \(( )\) to exclude them. In the given problem, we solve the inequality o explains that the solution set for o all values of \(z\) between o o . To write this in interval notation, we include both endpoints: o notati/o, the left endpoint is -26 and the right endpoint is 6, so our interval notation is o.oel
Solving Inequalities
To solve compound inequalities, we aim to isolate the variable from the middle term. Given the compound inequality oow, follow o.o, we subtract 1 from each part: \(-13 \leq \frac{1}{2}z \leq 3\). Finally, multiply by 2 to clear the fraction: \(-26 \leq z \leq 6\). Now, we successfully isolate \(z\) o to graph and write in interval notation.
Graphing Solutions
Graphing the solution set on a number line helps visually interpret the range of valid values for \(z\). For this problem, we plot \(-26\) and \(6\) with solid dots to indicate these values are included in the solution. Then, draw a line connecting these points.
- Solid dots at \(-26\) and \(6\)
- A line connecting these points
This effectively represents all values \(z\) between \(-26\) and \(6\).
Educational Algebra
Learning to solve compound inequalities is an essential algebra skill. These problems involve multiple steps such as isolating variables, handling fractions, and writing solutions in interval notation. It's like solving puzzles where each step brings you closer to the final picture.
Remember: always perform the same operation on all parts of the compound inequality and carefully check your work by plugging values back into the original inequality.
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