Problem 82 - A Graphical Approach to College Algebra

Question

Solve each equation and inequality analytically. Use interval notation to write the solution set for each inequality. (a) \(5-3 x=x+1\) (b) \(5-3 x \leq x+1\) (c) \(5-3 x \geq x+1\)

Step-by-Step Solution

Answer

(a) \(x = 1\); (b) \([1, \infty)\); (c) \((- fty, 1]\)

Step 1: Rearrange Equation (a)

Start by rearranging the given equation \(5-3x = x+1\). Move all the \(x\) terms to one side and constants to the other side, resulting in \(5 - x = 1 + 3x\).

Step 2: Simplify Equation (a)

Further simplify the equation from step 1 by subtracting \(x\) from both sides: \(5 = 4x + 1\). Then, subtract 1 from both sides to get \(4x = 4\).

Step 3: Solve for x in Equation (a)

Divide both sides by 4 to isolate \(x\): \(x = 1\). Therefore, the solution to equation (a) is \(x = 1\).

Step 4: Set Up Inequality (b)

For the inequality \(5 - 3x \leq x + 1\), start by subtracting \(x\) from both sides to isolate \(x\) terms on one side: \(5 - 3x - x \leq 1\).

Step 5: Simplify Inequality (b)

Simplify the inequality: \(5 - 4x \leq 1\). Now, subtract 5 from both sides to get \(-4x \leq -4\).

Step 6: Solve Inequality (b)

Divide both sides by \(-4\), remembering that dividing an inequality by a negative number reverses the inequality sign: \(x \geq 1\). The solution is \([1, \infty)\) in interval notation.

Step 7: Set Up Inequality (c)

For the inequality \(5 - 3x \geq x + 1\), start by subtracting \(x\) from both sides: \(5 - 3x - x \geq 1\).

Step 8: Simplify Inequality (c)

Simplify to get \(5 - 4x \geq 1\). Then subtract 5 from both sides to obtain \(-4x \geq -4\).

Step 9: Solve Inequality (c)

Divide both sides by \(-4\), remembering to reverse the inequality sign: \(x \leq 1\). The solution in interval notation is \((-fty, 1]\).

Key Concepts

interval notation

Interval notation is a simple way to write the set of all solutions for inequalities. It is particularly useful in college algebra, where you often encounter inequalities that you need to express clearly.
In interval notation, you use brackets to show whether endpoints are included or not:

• Square brackets \[ \] show that an endpoint is included in the interval.
• Parentheses ( ) indicate the endpoint is not included.

For example, if you have the solution \(x \geq 1\), you would write it in interval notation as \([1, \infty)\). This says that \(x\) can be any number greater than or equal to 1, and there is no upper limit.
Similarly, for a solution like \(x \leq 1\), the interval notation would be \((-\infty, 1]\). Here, \(x\) can take any value less than or equal to 1, extending indefinitely to the left on the number line.
Understanding interval notation helps make solutions more concise and easier to analyze, ensuring you communicate all possible solutions effectively.

analytical solution

Finding an analytical solution involves solving equations or inequalities using algebraic manipulations. It's a method where you directly calculate the answer, as opposed to using numerical methods or simulations.
In the exercises above, solving analytically involves:

• Rearranging terms to isolate variables.
• Performing operations such as addition, subtraction, multiplication, or division to simplify the equations or inequalities.
• Finding exact solutions rather than approximations.

For example, in equation (a), \(5 - 3x = x + 1\), you first rearrange terms to bring all \(x\) terms to one side and constants to the other. This eventually gives you \(4x = 4\). Dividing both sides by 4 yields the solution \(x = 1\).
Similarly, in inequality (b), \(5 - 3x \leq x + 1\), you rearrange and simplify to find \(-4x \leq -4\). When you divide by \(-4\), you flip the inequality sign, arriving at \(x \geq 1\).
This approach provides a clear pathway to determine exact and descriptive solutions using mathematical reasoning. The ability to solve problems analytically is a valuable skill in college algebra.

college algebra

College algebra serves as a foundation for higher-level math courses, equipping you with essential skills in algebraic manipulation and problem-solving.
A big part of college algebra includes understanding how to solve different types of equations and inequalities, like linear, quadratic, or rational ones. Here are some core elements:

• Equation Solving: You must be able to manipulate equations to find the unknown variable values efficiently.
• Inequalities: Knowing how to deal with inequalities, including understanding how operations like multiplication and division can change inequality signs, is crucial.
• Graphing: Being able to visualize equations and their solutions on a graph is another key component.

In our exercise, we tackled linear equations and inequalities by isolating variables and using interval notation for clarity.
These exercises are fundamental practice in college algebra, fostering a deeper understanding of how mathematical operations relate to real-world situations and putting those skills to test in more complex scenarios.