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Chapter 8: Problem 35

Solve. $$ |t|+1.1=6.6 $$

Short Answer

Expert verified
t = 5.5 or t = -5.5

Step by step solution

01

Isolate the absolute value expression

To start, subtract 1.1 from both sides of the equation to isolate the absolute value expression. \(|t| + 1.1 - 1.1 = 6.6 - 1.1\)This simplifies to:\(|t| = 5.5\)
02

Set up two separate equations

The absolute value equation \(|t| = 5.5\) means that \(t\) can be either positive or negative. This gives us two equations:\(t = 5.5\)and\(t = -5.5\)
03

Solve each equation

We now solve each of the equations from Step 2:For \(t = 5.5\), the solution is simply:\(t = 5.5\)For \(t = -5.5\), the solution is:\(t = -5.5\)
04

Write the final solution

Combining both solutions from Step 3, we get:\(t = 5.5\) or \(t = -5.5\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

absolute value
The concept of absolute value is essential in solving many kinds of equations. Absolute value represents the distance of a number from zero on the number line, regardless of its direction. That means the absolute value of both -5 and 5 is 5 because both are 5 units away from zero. We denote it using vertical bars, like \(|t|\).
In our example, \(|t| + 1.1 = 6.6\), the absolute value affects how we handle the equation. To solve it, we need to first isolate \(|t|\) on one side.
isolation of variables
Isolating variables is a crucial step in solving equations. It means moving terms around to get the variable by itself on one side of the equation.
In the given exercise, we isolate the absolute value expression first. The initial equation is: \( |t| + 1.1 = 6.6 \).
We subtract 1.1 from both sides to isolate \(|t|\), resulting in \(|t| = 5.5\). This makes it easier to work with the equation in the following steps. Remember: isolating the variable simplifies the problem and sets up the next steps for solving it.
linear equations
A linear equation is an equation between two variables that gives a straight line when plotted on a graph. However, in absolute value problems, we often deal with piecewise solutions.
For the equation \(|t| = 5.5\), the absolute value expression means there are two possible solutions: \( t = 5.5 \) and \( t = -5.5 \). These are derived from the property that absolute value measures distance, thus \(|t| = 5.5\) splits into \(t = 5.5\) and \(t = -5.5\).
Linear equations can be solved using basic algebraic techniques, and this exercise simply extended the property of absolute values to linear equations by creating two scenarios to solve.

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