Problem 21 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Write each solution set in interval notation. $$\frac{4 x+7}{-3} \leq 2 x+5$$

Step-by-Step Solution

Answer

The solution is \([-2.2, \infty)\).

Step 1: Move all terms to one side

To begin solving the inequality, move all terms involving the variable to one side of the inequality. Subtract \(2x + 5\) from both sides:\[ \frac{4x + 7}{-3} - (2x + 5) \leq 0 \]

Step 2: Combine like terms

First distribute and simplify the inequality. Note that when you divide by a negative number, the inequality direction changes: \[ \frac{4x + 7}{-3} - 2x - 5 \leq 0 \] becomes \[ \frac{4x + 7}{-3} - \frac{6x + 15}{3} \leq 0 \]. Multiplying the whole equation by \(-3\):\[ 4x + 7 + 6x + 15 \geq 0 \]Simplify to get:\[ 10x + 22 \geq 0 \]

Step 3: Solve for x

Solve the inequality for \(x\). Subtract 22 from both sides:\[ 10x \geq 14 \]. Then divide by 10:\[ x \geq -2.2 \]

Step 4: Write the solution in interval notation

The solution to the inequality in interval notation is \([-2.2, \infty)\). This means \(x\) can be any value greater than or equal to -2.2.

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real numbers. It is used to denote the solution sets of inequalities. When using interval notation, we use square brackets like \texttt{[ ]} whenever a value is included in the set, and parenthesis like \texttt{( )} when a value is not included. For example, the interval \([-2.2, \text{∞})\) means that \(x\) can be any number greater than or equal to -2.2 and extends to positive infinity:

• \texttt{[a, b]}: both a and b are included in the interval
• \texttt{(a, b]}: a is not included, but b is included
• \texttt{[a, b)}: a is included, but b is not included
• \texttt{(a, b)}: neither a nor b is included

For our solution \([-2.2, \text{∞})\), it indicates that -2.2 is part of the solution set because we use a square bracket next to -2.2. Infinity always uses parentheses because it is a concept rather than a number you can reach or include.

Linear Inequalities

Linear inequalities are algebraic expressions that use inequality signs instead of an equals sign. They express a relationship where one quantity is not necessarily equal to another but rather less than (\texttt{<}), less than or equal to (\texttt{≤}), greater than (\texttt{>}), or greater than or equal to (\texttt{≥}) another quantity. To solve linear inequalities:

• Move all variable terms to one side using addition or subtraction
• Simplify both sides, combining like terms
• Isolate the variable on one side using multiplication or division
• Note: Dividing or multiplying both sides by a negative number flips the inequality sign

For our original exercise, starting from the inequality \(\frac{4x + 7}{-3} ≤ 2x + 5\), we manipulate it by moving all terms to one side and simplifying. We combine the like terms and solve for the variable to get \(x ≥ -2.2\). Finally, we write this solution in interval notation.

Algebra

Algebra involves using symbols and letters to represent numbers in equations and expressions. It gives us tools to solve for unknowns by performing logical and arithmetic operations. Here are some key points:

• Variables represent unknown values
• Constants are fixed values
• Operators like \texttt{+}, \texttt{-}, \texttt{*}, and \texttt{/} help us perform basic arithmetic on variables and constants
• An equation is a mathematical statement with an equals sign indicating that two expressions are equivalent

For inequalities, rather than stating that two expressions are equal, they show the relative size of one expression to another. When solving the inequality \(\frac{4x + 7}{-3} ≤ 2x + 5\), algebraic manipulation involves moving terms, distributing, combining like terms, and finally isolating the variable. These steps help us to find the range of possible values for the variable and express them succinctly in interval notation.