Problem 31 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. See Example 3. $$ t+1-3 t \geq t-20 $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Inequality

Start by simplifying both sides of the inequality by combining like terms. The inequality is given as: \[ t + 1 - 3t \geq t - 20 \]Combine the \( t \) terms on the left side:\[ (t - 3t) + 1 \geq t - 20 \]This simplifies to:\[ -2t + 1 \geq t - 20 \]

Step 2: Move Terms to One Side

Move all terms involving \( t \) to one side and constant terms to the other. Start by subtracting \( t \) from both sides:\[ -2t - t + 1 \geq -20 \]This becomes:\[ -3t + 1 \geq -20 \]

Step 3: Isolate the Variable

Subtract 1 from both sides to isolate the \( t \)-terms:\[ -3t \geq -21 \]

Step 4: Solve for t

To solve for \( t \), divide each side of the inequality by \( -3 \). When dividing by a negative number, reverse the inequality symbol:\[ t \leq 7 \]

Step 5: Graph the Solution

To graph \( t \leq 7 \), draw a number line. Shade all numbers to the left of 7, including 7 itself, to represent all numbers less than or equal to 7.

Step 6: Write in Interval Notation

Write the solution set in interval notation. Since \( t \) can be any number less than or equal to 7, it is represented as:\[ (-\infty, 7] \]

Key Concepts

Graphing Inequalities

To visualize the solution of an inequality, we use a number line. The solution, for example, if expressed as \( t \leq 7 \), means all numbers less than or equal to 7. Start by placing a dot or circle at the value 7 on a drawn number line. Since the inequality includes the "equal to" part (\( \leq \)), you fill in the circle to indicate that 7 is included. If it were only "less than" (\( < \)), the circle would remain open to show exclusion. Once the point is marked, shade the entire area to the left of this point. This shading shows that all numbers less than 7 are included in the solution. Visual representation through graphing is very helpful in understanding how inequalities range over a set of numbers. Let's recap this visually simple process to make solving inequalities much clearer. Always remember to change the direction of the shading or the circle filling when dealing with a strict inequality!

Interval Notation

Interval notation is a shorthand used to express ranges of solutions for inequalities. It provides a concise way of presenting the solution set of an inequality like \( t \leq 7 \). In our scenario, the numbers that solve the inequality include every number less than or equal to 7, stretching back infinitely. In interval notation, you write this as \((-\infty, 7]\). The parentheses \((-\infty,\) indicate that infinity is not a specific number and can't be included, hence the use of a parenthesis. The square bracket after 7, \([7]\), shows that 7 is included in the solution. Interval notation helps to easily and neatly convey the solution set without the need to graph every time. It's a very common method used in algebra to summarize solutions efficiently.

Combining Like Terms

When you solve an inequality, often the first step involves simplifying by combining like terms. In the given inequality \( t + 1 - 3t \geq t - 20 \), the left side involves combining terms that have the same variable, \( t \). Specifically, you have two terms with the variable \( t \): \( t - 3t \). These can be combined by performing the subtraction to get \(-2t\). This yields a simpler inequality: \(-2t + 1 \geq t - 20\). Understanding how to combine like terms makes solving inequalities, or any algebraic expressions, more manageable. It allows you to streamline an expression by either reducing the number of terms or decreasing the expression's complexity. This is the foundational step for solving equations efficiently.