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Magazine Chapter 2: Problem 28
Solve each inequality. Graph the solution set and write it in interval notation. $$ 7(x-2)+x \leq-4(5-x)-12 $$
Short Answer
Expert verified
The solution is \( x \leq -4.5 \), written in interval notation as \((-\infty, -4.5]\).
Step by step solution
01
Distribute and Simplify
First, distribute the numbers outside the parentheses: \( 7(x-2) \) becomes \( 7x - 14 \) and \( -4(5-x) \) becomes \( -20 + 4x \). Substitute these values back into the inequality: \( 7x - 14 + x \leq -20 + 4x - 12 \). This simplifies to \( 8x - 14 \leq 4x - 32 \) after combining like terms on each side.
02
Isolate Variable Terms
Next, get all the \( x \) terms on one side by subtracting \( 4x \) from both sides: \( 8x - 4x - 14 \leq -32 \). Simplifying this gives \( 4x - 14 \leq -32 \).
03
Eliminate Constant Term
Add 14 to both sides to isolate the terms with \( x \): \( 4x - 14 + 14 \leq -32 + 14 \). This simplifies to \( 4x \leq -18 \).
04
Solve for the Variable
Now, divide all terms by 4 to solve for \( x \): \( \frac{4x}{4} \leq \frac{-18}{4} \). Simplifying gives \( x \leq -4.5 \).
05
Graph the Inequality
On a number line, plot a point at -4.5 and shade to the left, indicating all numbers less than or equal to -4.5 are part of the solution set. Use a closed circle at -4.5 because the value is included.
06
Write the Solution in Interval Notation
The solution in interval notation is \( (-\infty, -4.5] \). This represents all the numbers less than or equal to -4.5.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Interval Notation
Interval notation is a way to describe a set of numbers along the number line using intervals. It is especially useful when representing solutions to inequalities. Imagine it as a shorthand to show where a variable lies on the number line without graphing it.
Here are some key points about interval notation:
Here are some key points about interval notation:
- Use parentheses, "(", ")", for numbers that are not included in the solution. They signify an open interval.
- Use square brackets, "[", "]", for numbers that are included. These signify a closed interval.
- If an interval extends indefinitely to the left or right, use \( -\infty \) or \( \infty \). Important: infinity signs are always accompanied by parentheses, never brackets, because infinity isn't a number we can reach.
- Starting at \( -\infty \) (going infinitely left, which is why we use a parenthesis).
- Up to and including \( -4.5 \) (indicated by the bracket).
Graphing Inequalities
Graphing inequalities involves showing the set of values that satisfy an inequality on a number line. This is a visual method to understand what numbers are part of a solution set and is very useful for grasping inequalities intuitively.
Here鈥檚 how to graph an inequality like "\(x \leq -4.5\)":
Here鈥檚 how to graph an inequality like "\(x \leq -4.5\)":
- Draw a horizontal line to represent the number line.
- Locate the point \( -4.5 \) on the number line.
- Place a closed circle on \( -4.5\). A closed circle shows that \( -4.5\) is part of the solution set because of the "less than or equal to".
- Shade the line to the left of \( -4.5 \), indicating all values less than or equal to \( -4.5 \) are included.
Algebraic Manipulation
Algebraic manipulation is the technique used to transform equations or inequalities into a simpler or more understandable form. It involves a few strategic moves to isolate variables and solve equations seamlessly step by step.
In the Original Exercise, this process was crucial.
In the Original Exercise, this process was crucial.
- **First, distribute** numbers across parentheses: This involves multiplying the number outside the parentheses by every term inside. E.g., \( 7(x-2) \) becomes \( 7x - 14 \).
- **Combine like terms**: Gather all similar variable terms and constants on their respective sides.
- **Isolate the variable term**: You do this by moving terms to one side of the inequality, leaving the variable by itself on either the left or right.
- **Eliminate the constant**: Add or subtract constants to further isolate your variable term.
- **Solve for the variable**: Finally, divide or multiply to solve for the variable, achieving a simple inequality like \( x \leq -4.5 \).
