Problem 86 - A Graphical Approach to Precalculus with Limits

Question

Solve each inequality analytically, writing the solution set in interval notation. Support your answer graphically. (Hint: Once part (a) is done, the answer to part (b) follows.) (a) \(5-3 x \leq-11+x\) (b) \(5-3 x>-11+x\)

Step-by-Step Solution

Answer

(a) \([4, \infty)\); (b) \((-\infty, 4)\)

Step 1: Simplify the Inequality for (a)

For (a), start with the inequality: \[5 - 3x \leq -11 + x\]Add \(3x\) to both sides to eliminate \(-3x\) from the left-hand side:\[5 \leq -11 + 4x\]Then, add 11 to both sides to isolate the term with \(x\) on one side:\[16 \leq 4x\]

Step 2: Solve for x in Inequality (a)

Continue from the inequality \(16 \leq 4x\). Divide both sides by 4 to solve for \(x\):\[4 \leq x\]This can also be written as:\[x \geq 4\]

Step 3: Write Solution for (a) in Interval Notation

Since \(x\) must be greater than or equal to 4, the solution in interval notation is:\[ [4, \infty) \]

Step 4: Simplify the Inequality for (b)

For (b), start with the inequality:\[5 - 3x > -11 + x\]Again, add \(3x\) to both sides to eliminate \(-3x\) from the left-hand side:\[5 > -11 + 4x\]Then, add 11 to both sides:\[16 > 4x\]

Step 5: Solve for x in Inequality (b)

Continue from the inequality \(16 > 4x\). Divide both sides by 4 to isolate \(x\):\[4 > x\]This can also be written as:\[x < 4\]

Step 6: Write Solution for (b) in Interval Notation

Since \(x\) must be less than 4, the solution in interval notation is:\[ (-\infty, 4) \]

Key Concepts

Interval Notation

Interval notation is a mathematical shorthand used to express the set of solutions we find for inequalities. Rather than listing individual numbers, interval notation provides a concise way to communicate a continuous range of values. For example, when you see \([4, \infty)\), it means that the solution set includes all numbers from 4 to infinity, with 4 included because of the bracket "[". \ To help:

• A square bracket [ or ] means the endpoint is included in the set.
• A parenthesis ( or ) indicates the endpoint is not included.

For the exercise given:

• The inequality \(5 - 3x \leq -11 + x\) resolves such that \(x \geq 4\), leading to the interval \([4, \infty)\).
• While \(5 - 3x > -11 + x\) resolves to \(x < 4\), which is expressed as \((-\infty, 4)\).

Understanding interval notation can make interpreting and solving inequalities much more intuitive.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to find the value of variables that make the inequality true. In the given exercise, we were tasked to solve inequalities by isolating the variable:

• Combine like terms to simplify expressions.
• Add or subtract on both sides to keep the inequality balanced.
• Multiply or divide all terms by the same non-zero number.

For instance, in solving \(5 - 3x \leq -11 + x\), we first added 3x to both sides to remove it from the left. Our aim was to have all variables on one side. Next, we added 11 to both sides, aiming to have constants on the opposite side. This gave us \(16 \leq 4x\), allowing us to find that \((x \geq 4)\) when divided by 4.For the second inequality \(5 - 3x > -11 + x\), similar steps allowed us to find that \(x < 4\).Recognizing these steps in algebraic manipulation lets us solve inequalities efficiently and accurately.

Graphical Representation

Graphical representation offers a visual understanding of inequalities. This means plotting the solution set on a number line or graph to visually depict what the solution looks like. Plotting helps to quickly see which values satisfy the inequality without having to ruminate through every single number.Here are steps to plot a simple inequality like those in the exercises:

• Choose a number line.
• For \(x \geq 4\), place a filled circle at 4 and shade the line to the right, indicating all numbers equal to or greater than 4.
• For \(x < 4\), draw an open circle at 4 and shade the line to the left, demonstrating all numbers less than 4.

These visual tools are quick checks for solutions found analytically and are crucial for learning and confirming the set of possible values that satisfy the inequalities. By continually using both methods—analytical and graphical—confidence in solving inequalities will steadily build.