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Magazine Chapter 2: Problem 76
Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ 3(3-x) \geq 6+x $$
Short Answer
Expert verified
The solution in interval notation is \((-\infty, \frac{3}{4}]\).
Step by step solution
01
Distribute and Simplify
First, distribute the \(3\) across the terms inside the parentheses on the left-hand side: \(3(3-x) = 9 - 3x\). This transforms the inequality to: \(9 - 3x \geq 6 + x\).
02
Isolate the Variable Term
Next, get all terms containing \(x\) on one side of the inequality. Adding \(3x\) to both sides results in: \(9 \geq 6 + x + 3x\). Simplifying gives: \(9 \geq 6 + 4x\).
03
Solve for \(x\)
Subtract \(6\) from both sides to isolate \(4x\): \(9 - 6 \geq 4x\), which simplifies to \(3 \geq 4x\). Divide both sides by \(4\) to solve for \(x\): \(\frac{3}{4} \geq x\) or equivalently \(x \leq \frac{3}{4}\).
04
Write the Solution in Interval Notation
The solution \(x \leq \frac{3}{4}\) in interval notation is written as \(( -\infty, \frac{3}{4} ]\). This notation includes all numbers less than or equal to \(\frac{3}{4}\).
05
Graph the Solution on a Number Line
Draw a number line and place a closed dot at \(\frac{3}{4}\) to indicate that \(x\) can be equal to \(\frac{3}{4}\). Shade the region extending from \(\frac{3}{4}\) to the left toward \(-\infty\), representing all numbers less than or equal to \(\frac{3}{4}\).
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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 method of representing sets of numbers along a number line. It's concise and helps communicate the range of solutions in inequalities.
In interval notation, brackets and parentheses are used to indicate whether endpoints are included in the interval:
In interval notation, brackets and parentheses are used to indicate whether endpoints are included in the interval:
- Round parenthesis ( ) means the endpoint is not included. For example, (3,5) includes all numbers between 3 and 5, but not the numbers 3 and 5 themselves.
- Square bracket [ ] means the endpoint is included. For example, [3,5] includes the numbers 3 and 5 along with all numbers in between.
Number Line Graphing
Graphing on a number line provides a visual way to understand the solution of inequalities. It's particularly helpful for comprehending what values an inequality allows or restricts.
To graph x 鈮� 3/4:
To graph x 鈮� 3/4:
- First, draw a horizontal line to represent the number line.
- Locate the position on the line where 3/4 would fall. You can imagine dividing the space between 0 and 1 into four parts.
- Place a closed dot over this point. The closed dot shows that 3/4 is part of the solution.
- Shade to the left of the closed dot all the way to -鈭� to indicate that all numbers less than 3/4 are included in the solution.
Variable Isolation
Isolating the variable in an equation or inequality means reorganizing the expression so that the variable stands alone on one side. It's a critical skill when solving for a variable's value, especially in inequalities where the variable鈥檚 position is key.
To isolate a variable, you may need to:
To isolate a variable, you may need to:
- Add, subtract, multiply, or divide both sides of the inequality by a number.
- Remember that if you multiply or divide by a negative number, you need to flip the inequality sign.
- Then, by adding 3x to both sides: 9 鈮� 6 + 4x.
- Next, subtracting 6 from both sides gives: 3 鈮� 4x.
- Finally, dividing both sides by 4 results in: x 鈮� 3/4.
Compound Inequalities
Compound inequalities involve more than one inequality combined together, expressing a range of values meeting several conditions simultaneously. They are pivotal when solving real-world problems requiring two or more conditions.
Compound inequalities can be connected by words like "and" or "or":
Compound inequalities can be connected by words like "and" or "or":
- "And" inequalities constrain solutions to be true for all parts. For example, x > 2 and x < 5 implies x is any value between 2 and 5, not including 2 and 5.
- "Or" inequalities include solutions satisfying any condition. For example, x < 1 or x > 8 implies x can be any number less than 1 or greater than 8.
