Problem 98 - College Algebra Essentials

Question

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y=2 x-11+3(x+2)\) and \(y\) is at most 0.

Step-by-Step Solution

Answer

\((- \infty , 1]\)

Step 1: Combine Like Terms

Combine like terms in the equation: \(2x + 3x +(-11 + 6) \leq 0\). This simplifies to \(5x - 5 \leq 0\).

Step 2: Solve For The Variable

Now, add 5 to both sides to isolate \(5x\) on the left side of the equation, resulting in: \(5x \leq 5\). Divide by 5 to both sides to solve for \(x\), which gives us \(x \leq 1\).

Step 3: Represent Solution in Interval Notation

The solution to the inequality in interval notation is \((- \infty , 1]\), where \(\infty\) represents infinity and the square bracket means the number 1 is included in the interval.

Key Concepts

Inequality Solving

When solving inequalities, we are looking for the set of all possible values of a variable that satisfy the given condition. Solving an inequality is similar to solving an equation, but with a few key differences.

• Just like equations, we can perform operations like addition, subtraction, multiplication, or division on both sides of an inequality.
• Be careful when multiplying or dividing both sides by a negative number, as this flips the inequality sign.

Inequalities are often represented on a number line, which helps visualize the solutions. In interval notation, we use brackets and parentheses to denote whether endpoints are included (square brackets) or excluded (parentheses). For example, a solution like \(x \leq 1\) is represented as \((-\infty, 1]\), meaning all numbers up to and including 1 are part of the solution.
Understanding inequalities is useful in many areas of mathematics and everyday problem solving. It prepares you for more advanced topics dealing with ranges of values.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \(x\) or \(y\)), and operations (like addition, subtraction, multiplication, or division). An expression like \(y = 2x - 11 + 3(x + 2)\) involves both constants and variables.
To work with algebraic expressions, you need to perform operations and simplify them as much as possible. Simplification may involve removing parentheses, combining like terms, or factoring.
In the original exercise, we first needed to eliminate parentheses and then combine similar terms, which means terms with the same variable part:

• \(2x\)
• \(3x\)

By performing these operations, algebraic expressions become easier to work with and solve for the desired variables. Mastering algebraic expressions is essential in problem-solving, allowing us to model real-life situations effectively.

Combining Like Terms

Combining like terms is a process used to simplify algebraic expressions and solve equations or inequalities. Like terms are terms that have the same variable raised to the same power. For example, \(2x\) and \(3x\) are like terms, because they both contain \(x\) raised to the first power.
To combine like terms, simply add or subtract their coefficients. This reduces the number of terms and makes the expression simpler.

• In the exercise, we combined \(2x\) and \(3x\) to get \(5x\).
• We also simplified the constant terms: \(-11 + 6\) which results in \(-5\).

After combining, our expression \(2x + 3x + (-11 + 6)\) became \(5x - 5\).
Understanding how to combine like terms efficiently is key to simplifying expressions and solving more complex algebraic problems.