Problem 35 - 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 \(x \geq 5\). \(-10 \leq 2+4 x\)

Step-by-Step Solution

Answer

The interval is expressed as \(x \geq -3\).

Step 1: Identify the Inequality

We are given the interval \(-10 \leq 2 + 4x\). Our task is to solve this inequality to express it in terms of \(x\).

Step 2: Isolate the term containing x

Subtract 2 from both sides of the inequality to isolate the term containing \(x\). This gives us \(-10 - 2 \leq 4x\), which simplifies to \(-12 \leq 4x\).

Step 3: Solve for x

To solve for \(x\), divide both sides of the inequality \(-12 \leq 4x\) by 4. This results in \(-3 \leq x\).

Step 4: Final Expression

Thus, the interval \(-10 \leq 2 + 4x\) can be expressed as the inequality \(x \geq -3\).

Key Concepts

Understanding Intervals

Intervals are a way to express a range of values, showing where a variable can belong. In mathematics, intervals are often represented using different notations. To convert these notations into inequalities can be crucial in understanding the set of possible solutions.

There are different types of intervals:

• Closed intervals are denoted with square brackets like \([-10, \ \infty)\). It includes the endpoint(s) of the interval.
• Open intervals use parentheses like \( (5,10) \). It excludes the endpoints.
• Half-open intervals, such as \( [5,10) \), include one endpoint but not the other.
• Infinite intervals, like \( (-\infty, 5] \), extend indefinitely in one direction.

To write intervals as inequalities, follow the bounds suggested by the endpoints and understand whether those endpoints are included or excluded.

Basics of Algebra in Inequalities

Algebra forms the basis for many concepts in mathematics, including solving inequalities. It involves operations such as addition, subtraction, multiplication, and division of variables and constants. By using algebra, we can rearrange and simplify expressions to uncover the values for unknowns.

When applying algebra in the context of inequalities, the process is similar to regular equations with a notable exception. The inequality sign maintains the relationship direction unless both sides of the inequality are multiplied or divided by a negative number, at which point the direction of the inequality sign flips.

For example, let's break down how algebra is applied:

• Identify the inequality you need to solve, e.g., \(-10 \leq 2 + 4x\).
• Use subtraction to simplify: subtract 2 from both sides, leading to \(-12 \leq 4x\).
• Divide by 4, yielding \(-3 \leq x\).

This algebraic process helps us change a complicated expression into something more manageable.

Solving Inequalities Step by Step

When solving inequalities, the aim is to find out which values of a variable satisfy the inequality. You need a strategy that often involves restructuring the inequality until the variable is isolated.

Here’s a simple step-by-step guide to solving inequalities, similar to the example provided:

• Step 1: Isolate terms.
  Start with the given inequality. Rearrange terms such that the variable of interest is on one side alone. In our example, this involved subtracting 2 from both sides.
• Step 2: Simplify.
  Perform the necessary operations (like subtraction or addition) to simplify the inequality. This led us to \(-12 \leq 4x\).
• Step 3: Solve for the Variable.
  Divide or multiply to solve for the variable, keeping in mind that dividing or multiplying by a negative flips the inequality. Here, dividing by 4 provided us \(-3 \leq x\).
• Step 4: Write in standard inequality format.
  Express the solution clearly, highlighting the range of possible values for the variable.

With these steps, solving inequalities becomes more straightforward, giving you the interval of solutions in inequality form.