Problem 65 - Elementary and Intermediate Algebra

Question

Solve each inequality. Write the solution set in interval notation and then graph it. $$ t+1-3 t \geq t-20 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, 7]\).

Step 1: 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 \( -2t + 1 \geq t - 20 \).

Step 2: 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 \( -3t + 1 \geq -20 \).

Step 3: Isolate the Variable

Subtract 1 from both sides to begin isolating \( t \): \( -3t + 1 - 1 \geq -20 - 1 \). This simplifies to \( -3t \geq -21 \).

Step 4: Solve for the Variable

Divide both sides by \( -3 \), remembering that dividing by a negative number flips the inequality sign: \( t \leq 7 \).

Step 5: Write Solution in Interval Notation

The solution \( t \leq 7 \) in interval notation is \( (-\infty, 7] \).

Step 6: 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.

Key Concepts

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:

• 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.

For example, the solution to the inequality \(t \leq 7\) is written as \((-\infty, 7]\) because \(t\) includes all numbers less than or equal to 7. The parentheses used around \(-\infty\) reflect that infinity is not a number we can reach, so it can never be included in the interval.

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:

• 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.

Using number lines for graphical representation not only supports algebraic solutions but also strengthens understanding through visual confirmation. It becomes easier to see the range of solutions by looking at the shaded area.

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 \(t + 1 - 3t \geq t - 20\):

• 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 \(t \leq 7\).

It's crucial to remember this rule about reversing the inequality when multiplying or dividing by negative numbers. This keeps your solution logical and correct, providing accurate representations of numerical relationships.