Problem 15 - Thomas Calculus

Question

Solve the equations in Exercises \(13-18\) $$ |2 t+5|=4 $$

Step-by-Step Solution

Answer

The solutions are \(t = -\frac{1}{2}\) and \(t = -\frac{9}{2}\).

Step 1: Understand Absolute Value Equation

The equation given is \(|2t + 5| = 4\). An absolute value \(|x|\) represents the distance of \(x\) from zero on a number line. Thus, it can be either \(x = 4\) or \(x = -4\).

Step 2: Set Up Two Separate Equations

Since \(|2t + 5| = 4\), this implies two possibilities: 1. \(2t + 5 = 4\) 2. \(2t + 5 = -4\).

Step 3: Solve the First Equation

Start with the first possibility, \(2t + 5 = 4\). Subtract 5 from both sides: \[2t = 4 - 5\]\[2t = -1\]Divide both sides by 2:\[t = -\frac{1}{2}\].

Step 4: Solve the Second Equation

Now solve the second possibility, \(2t + 5 = -4\).Subtract 5 from both sides:\[2t = -4 - 5\]\[2t = -9\]Divide both sides by 2:\[t = -\frac{9}{2}\].

Step 5: Write General Solution

The solutions for \(|2t + 5| = 4\) are \(t = -\frac{1}{2}\) and \(t = -\frac{9}{2}\).

Key Concepts

Solving Absolute Value Equations

When we encounter an absolute value equation like \(|2t + 5| = 4\), it signifies the distance on the number line. The absolute value takes into account both the positive and negative scenarios of the number. This is why we end up with two separate equations to solve - one for the positive case and one for the negative case. To break it down further, if we have \(|x| = a\), then the solutions can be expressed as \(x = a\) or \(x = -a\). This is because the absolute value represents a measure without direction, so both scenarios are valid solutions. In our example:

• Setting the inside of the absolute value to the positive, \(2t + 5 = 4\).
• Setting to the negative, \(2t + 5 = -4\).

Using this method, we can solve and find all possible values for the variable in an absolute value equation.

Absolute Value Properties

Absolute value properties are helpful when tackling equations that incorporate \(|x|\). The absolute value of any number \(x\) denoted as \(|x|\) is always non-negative. Certain notable properties include:

• Non-Negativity: \(|x| \geq 0\) for any \(x\). That means the result of an absolute value is never negative.
• Identity Property: If \(|x| = 0\), then \(x = 0\). This makes sense because zero has no distance from itself on the number line.
• Symmetry: \(|-x| = |x|\). Negative values become positive, thus symmetrical around zero.
• Triangle Inequality: \(|x + y| \leq |x| + |y|\), which helps us understand how sums of absolute values work compared to the absolute value of a sum.

These properties facilitate an understanding of why absolute value equations require two solutions and help in comprehending their application in real-world scenarios.

Linear Equations Solutions

Solving the linear equations is the next step after setting up the equation through absolute value understanding. In our case, once we have the equations \(2t + 5 = 4\) and \(2t + 5 = -4\), they need to be solved individually to find the values of \(t\).Let's consider the first equation:

• Subtract 5 from both sides which gives \(2t = -1\).
• Divide by 2 which simplifies to \(t = -\frac{1}{2}\).

For the second equation, repeat the steps:

• Subtract 5 to end up with \(2t = -9\).
• Again, divide by 2 arriving at \(t = -\frac{9}{2}\).

Therefore, for an absolute value equation, after transforming it into linear equations, performing basic algebraic isolation gives all possible solutions for the variable in question. It’s a straightforward process of isolating the variable through addition, subtraction, multiplication, or division.