Problem 36 - Algebra for College Students

Question

Express each interval as an inequalit using the variable \(x\). For example, we can express the inter val \([5, \infty)\) as \(x \geq 5\). \(-9 \leq 1+2 x\)

Step-by-Step Solution

Answer

\(x \geq -5\)

Step 1: Write the Given Interval as an Inequality

We already have the inequality provided in the exercise: \(-9 \leq 1+2x\). This means that the expression \(1+2x\) should be greater than or equal to \(-9\).

Step 2: Isolate the Variable Term

Subtract 1 from both sides of the inequality to begin isolating the variable term \(2x\):\[-9 - 1 \leq 1 + 2x - 1\]This simplifies to \[-10 \leq 2x\].

Step 3: Solve for x

Divide each side of the inequality by 2 to solve for \(x\):\[\frac{-10}{2} \leq \frac{2x}{2}\]This simplifies to \[-5 \leq x \], which can be rewritten as \[x \geq -5\].

Key Concepts

Interval Notation

Interval notation is a shorthand way of writing sets of numbers, often used to describe a range of values that a variable like \(x\) can take. When writing intervals, we use brackets and parentheses to indicate whether the endpoints of the interval are included or excluded.

To help understand this notation better, remember:

• Square brackets \([ ]\) denote inclusive intervals, meaning the endpoint is included. For example, \([3, 7]\) includes both 3 and 7.
• Round parentheses \(( )\) denote exclusive intervals, meaning the endpoint is not included. For example, \((3, 7)\) means 3 and 7 are not part of the interval.
• If the interval extends to infinity, use round brackets because infinity is not a specific number and can't be included. Thus, \((5, \infty)\) describes numbers greater than 5.

In interval notation, an inequality like \(x \geq -5\) can be written as \([-5, \infty)\), indicating that \(x\) is any number greater than or equal to \(-5\). Learning interval notation makes it easier to express and comprehend mathematical ideas concisely.

Algebraic Manipulation

Algebraic manipulation involves the rearrangement of equations and inequalities to solve for a specific variable. The core idea is applying mathematical operations equally on both sides to maintain balance.

Consider why algebraic manipulation is useful:

• It helps isolate a variable, making it easier to solve an equation or inequality.
• Simplifies complex expressions to reveal the solution more clearly.
• Maintains the proper order of operations to ensure accurate results.

In our exercise, we started with \(-9 \leq 1 + 2x\). Subtracting 1 from both sides to get \(-10 \leq 2x\), we're one step towards isolating \(x\). Then we divide by 2 to solve for \(x\), resulting in \(-5 \leq x\) or \(x \geq -5\). Each step is an algebraic manipulation that simplifies the problem closer to a usable solution. Practicing these steps helps understand the balance needed in algebraic processes.

Inequalities in Algebra

Inequalities are a fundamental part of algebra, representing the relationship between expressions that are not necessarily equal. Keywords like "greater than", "less than", and their equal counterparts describe these relationships.

Essential rules when dealing with inequalities:

• When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips direction. For instance, if you divide \(-2x < 4\) by \(-2\), it becomes \(x > -2\).
• Performing the same operation on both sides maintains the inequality's validity, similar to balancing an equation.
• Graphing inequalities on a number line highlights the range of solutions visually.

Our example, \(-9 \leq 1 + 2x\), translated to \(-5 \leq x\) upon completion, demonstrates using inequalities to describe potential values for \(x\). Recognizing how inequalities function in algebra lends insight into real-world situations where exact values are unknown, ensuring a comprehensive problem-solving approach.