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Magazine Chapter 8: Problem 65
Solve each inequality. Write the solution set in interval notation and then graph it. $$ t+1-3 t \geq t-20 $$
Short Answer
Expert verified
The solution is \((-\infty, 7]\).
Step by step solution
01
Simplify the Inequality
Start by simplifying both sides of the inequality. Combine the terms on the left-hand side: \( t + 1 - 3t \). This simplifies to \( -2t + 1 \). So the inequality becomes \( -2t + 1 \geq t - 20 \).
02
Move Variable Terms to One Side
To isolate the variable terms, subtract \( t \) from both sides of the inequality: \( -2t + 1 - t \geq t - 20 - t \). This simplifies to \( -3t + 1 \geq -20 \).
03
Isolate the Variable
Subtract 1 from both sides to begin isolating \( t \): \( -3t + 1 - 1 \geq -20 - 1 \). This simplifies to \( -3t \geq -21 \).
04
Solve for the Variable
Divide both sides by \( -3 \), remembering that dividing by a negative number flips the inequality sign: \( t \leq 7 \).
05
Write Solution in Interval Notation
The solution \( t \leq 7 \) in interval notation is \( (-\infty, 7] \).
06
Graph the Solution
To graph \( t \leq 7 \), draw a number line. Shade the line to the left of 7 and include a closed circle at 7 to show that 7 is included in the solution set.
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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 of writing subsets of the real numbers. It is particularly useful for expressing the solution sets of inequalities. For instance, when we solve an inequality and find a range of values that satisfy it, we can use interval notation to denote this range.
There are two types of intervals:
There are two types of intervals:
- Closed intervals: These include the end numbers in the set, indicated by square brackets \[ [a, b] \]: It represents all numbers from \(a\) to \(b\), including both \(a\) and \(b\).
- Open intervals: These exclude the end numbers, denoted by parentheses like \((a, b)\): It includes all numbers between \(a\) and \(b\), but not \(a\) and \(b\) themselves.
Graphing Inequalities
Graphing inequalities on a number line visually represents the solution set. This helps students to clearly understand which numbers satisfy the inequality.
For graphing an inequality like \(t \leq 7\), follow these steps:
For graphing an inequality like \(t \leq 7\), follow these steps:
- Draw a number line: Make a horizontal line and mark key points that are significant to the inequality.
- Identify the critical value: In this case, 7 is the critical value. Use a solid dot at 7 because \(t \leq 7\) includes the number 7 itself.
- Shade the correct side: Since \(t\) is less than or equal to 7, shade the number line to the left of 7. This shows all the numbers less than or equal to 7 are part of the solution set.
Solving Algebraic Inequalities
Solving algebraic inequalities follows procedures similar to solving algebraic equations, but with a key difference: manipulating inequalities can change the sense (direction) of the inequality sign.
Here's a step-by-step approach to solving \(t + 1 - 3t \geq t - 20\):
Here's a step-by-step approach to solving \(t + 1 - 3t \geq t - 20\):
- Simplify both sides: Combine like terms where possible. Here, the left side simplifies to \(-2t + 1\).
- Bring variables together: Move all terms involving the variable to one side, resulting in \(-3t + 1 \geq -20\).
- Isolate the variable: Subtract constants from both sides to shift them away from the variable, simplifying to \(-3t \geq -21\).
- Divide by the coefficient: Divide every term by the coefficient of the variable. If this coefficient is negative, reverse the inequality sign. Dividing by \(-3\) results in \(t \leq 7\).
