Problem 51
Question
The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set for equations in set notation and use interval notation for inequalities. $$3=|4 t+5|$$
Step-by-Step Solution
Verified Answer
The short answer to the given absolute value equation is:
The solution set is \(\{-0.5, -2\}\).
1Step 1: Rewrite the equation as two linear equations
Since we are given an absolute value equation, we can write it as two linear equations:
\(3 = 4t + 5\) and \(-3 = 4t + 5\)
Now we will solve both equations separately.
2Step 2: Solve the first linear equation
Solve the equation for t:
\(3 = 4t + 5\)
Subtract 5 from both sides:
\(-2 = 4t\)
Divide both sides by 4:
\(-\frac{1}{2} = t\)
The first solution for t is -0.5.
3Step 3: Solve the second linear equation
Now, we solve the second equation:
\(-3 = 4t + 5\)
Subtract 5 from both sides:
\(-8 = 4t\)
Divide both sides by 4:
\(-2 = t\)
The second solution for t is -2.
4Step 4: Write the solution set in set notation
Since we have found two solutions for t, -0.5 and -2, we can write the solution set as follows:
\(\{-0.5, -2\}\)
The solution set for the given absolute value equation is \(\{-0.5, -2\}\).
Key Concepts
Linear EquationsSet NotationEquation SolvingInterval Notation
Linear Equations
Linear equations are fundamental to understanding algebra. They are equations of the first degree, which means that the highest power of the variable (usually denoted as \( t \)) is 1. This results in a graph that is a straight line. In the exercise given, linear equations arise from rewriting the initial absolute value equation into two separate linear equations. When solving these linear equations, we apply basic algebraic manipulations to isolate the variable and find its value. Key steps involve addition, subtraction, multiplication, or division to both sides of the equation until the variable stands alone on one side.
Linear equations are vital because they model many real-world situations and form the basis for more complex mathematical concepts.
Linear equations are vital because they model many real-world situations and form the basis for more complex mathematical concepts.
Set Notation
Set notation is a way to describe a collection of objects, which, in the context of the problem, are the solutions to the equation. It's a mathematical shorthand that describes elements that belong together. For instance, when we solve the absolute value equation and find solutions for \( t \), we express these solutions in set notation.
In our example, the solutions \(-0.5\) and \(-2\) are placed inside curly braces to denote a set:
In our example, the solutions \(-0.5\) and \(-2\) are placed inside curly braces to denote a set:
- \( \{-0.5, -2\} \)
Equation Solving
Solving equations involves finding the value of the variable that makes the equation true. This often means manipulating the equation through various algebraic methods, such as adding or subtracting terms on both sides, or dividing to isolate the variable. In absolute value equations, like the one given, this process begins by understanding that the absolute value \(|x|\) can equal a positive and a negative number. This leads us to split the absolute value equation into two separate linear equations.
The goal is to simplify each equation until you find a straightforward solution for the variable. The process requires consistent application of algebraic rules and a solid understanding of operations and inverses. Mastery of these skills improves problem-solving efficiency and accuracy.
The goal is to simplify each equation until you find a straightforward solution for the variable. The process requires consistent application of algebraic rules and a solid understanding of operations and inverses. Mastery of these skills improves problem-solving efficiency and accuracy.
Interval Notation
Interval notation is a method of indicating the set of solutions for inequalities. Though not used in absolute value equations directly, it becomes essential when dealing with inequalities related to them. This notation describes the intervals on the real-number line between specific numbers.
Intervals can be open (not including end values) or closed (including end values). For example, if we had an inequality with solutions between \(-2\) and \(3\), we would write it using interval notation as \(( -2, 3 )\) for an open interval or \([ -2, 3 ]\) for a closed interval.
Intervals can be open (not including end values) or closed (including end values). For example, if we had an inequality with solutions between \(-2\) and \(3\), we would write it using interval notation as \(( -2, 3 )\) for an open interval or \([ -2, 3 ]\) for a closed interval.
- Open intervals are indicated by parentheses: \((a, b)\)
- Closed intervals use brackets: \([a, b]\)
Other exercises in this chapter
Problem 50
The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set
View solution Problem 50
Graph each compound inequality. \(5 x-3 y>9\) and \(2 x+3 y \leq 12\)
View solution Problem 51
Graph each compound inequality. \(y \leq-x-1\) or \(x \geq 6\)
View solution Problem 52
The following exercises contain absolute value equations, linear inequalities, and both types of absolute value inequalities. Solve each. Write the solution set
View solution