Problem 97 - College Algebra Essentials

Question

Use interval notation to represent all values of \(x\) satisfying the given conditions. \(y=1-(x+3)+2 x\) and \(y\) is at least 4.

Step-by-Step Solution

Answer

The values of \(x\) satisfying the conditions are represented as \([3, \infty)\) in interval notation.

Step 1: Simplify the Expression

Simplify the expression \(y=1-(x+3)+2x\). Distribute the minus sign through the parenthesis and combine like terms. So, it simplifies to \(y=2x-2\).

Step 2: Solve the Inequality

Next, solve the inequality \(2x-2 \geq 4\). Start by adding 2 to both sides to isolate the term containing \(x\). This gives \(2x \geq 6\). Then, divide both sides by 2 to solve for \(x\). This results in \(x \geq 3\).

Step 3: Write the Solution in Interval Notation

The solution to the inequality is \(x \geq 3\). In interval notation, this is written as \([3, \infty)\). The square bracket indicates that the value of 3 is included in the solution.

Key Concepts

Solving Inequalities

Solving inequalities involves finding all possible values of a variable that satisfy the condition specified by the inequality. When solving inequalities, we often follow steps similar to solving equations, except we need to be cautious about direction changes when multiplying or dividing by negative numbers. In this exercise, after simplifying the expression and setting it greater than or equal to 4, we are tasked with solving the inequality \(2x - 2 \geq 4\).
Here are the steps involved:

• Isolate the variable term: Add 2 to both sides to remove the constant term on the left, resulting in \(2x \geq 6\).
• Solve for the variable: Divide each side by 2 to find \(x \geq 3\).

Once we have isolated \(x\), we can see that all values equal to or greater than 3 satisfy the inequality. Inequalities are essential not just in mathematics but also in real-world scenarios, such as determining which values are within a permitted range.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its simplest form. It makes equations easier to solve and understand by eliminating redundant parts and combining like terms. Let's look at how the expression \(y = 1 - (x + 3) + 2x\) was simplified:- First, apply the distributive property to remove the parenthesis, changing the expression within by distributing the "-" sign, which turns it into \(1 - x - 3\).- Next, arrange all like terms: Combine \(-x\) and \(2x\) to get \(1 + x - 3\).
Further combining results in: \(2x - 2\) – this is the expression's simplest form.When simplifying, always remember to:

• Apply distributive property properly: Ensure that all terms in parentheses are correctly affected by any factors or signs outside of them.
• Combine like terms: This means summing or subtracting coefficients of similar variables to further simplify the expression.

Having a simplified expression not only assists in solving problems but also minimizes the potential for errors in subsequent steps.

Combining Like Terms

Combining like terms is a fundamental skill in algebra used to simplify expressions and solve equations efficiently. It involves grouping together terms in an equation that have similar variables raised to the same power. In our exercise, the expression \(1 - (x + 3) + 2x\) required identifying and combining these like terms:

• Terms involving \(x\) were \(-x\) and \(2x\). When these were combined, they resulted in \(2x - x = x\).
• Numeric terms were \(1\) and \(-3\), which combined to form \(-2\).

Together, these steps transformed our initial expression into the simplified form \(2x - 2\).When combining like terms, always:

• Identify like terms: Look for terms with the same variable component, like \(x\), which can be added or subtracted from one another.
• Combine by performing arithmetic: Add or subtract the coefficients while keeping the variable part unchanged.

This simplification prepares the equation for solving and helps in clearly determining the relationships between various aspects of the equation.