Problem 27 - 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 \(x \geq 5\). \(-7 x-3 \leq 4\)

Step-by-Step Solution

Answer

The interval is \([-1, \infty)\).

Step 1: Start with the Inequality

We begin with the given inequality: \(-7x - 3 \leq 4\).

Step 2: Isolate the Variable Term

To isolate the term with \(x\), add 3 to both sides.\(-7x - 3 + 3 \leq 4 + 3\)This simplifies to:\(-7x \leq 7\).

Step 3: Solve for the Variable

To solve for \(x\), divide both sides by -7. Remember that dividing by a negative number reverses the inequality sign.\(x \geq -1\).

Step 4: Express as an Interval

Now, we can express \(x \geq -1\) as an interval in the form \([-1, \infty)\).

Key Concepts

Interval notation

Interval notation is a way of representing a set of numbers. It’s especially useful for describing intervals on the real number line. In this context, an interval is a range of numbers that includes all the numbers within two end points. Let's break down the types and specifics.

• Closed intervals: Denoted with square brackets, such as \[a, b\], and they include the endpoints \('a' and 'b'\).
• Open intervals: Represented with parentheses, such as (a, b), indicating the endpoints are not included.
• Infinite intervals: These extend indefinitely in one direction, for example, [a,∞) or (-∞,b].

For example, if we have an inequality like \(x \geq -1\), it translates to the interval [-1, ∞), showing that \(x\) includes -1 and all larger numbers. Understanding interval notation makes it easier to comprehend and express solutions to inequalities.

Solving inequalities

Solving inequalities involves figuring out the values of the variable that satisfy the inequality condition, much like solving equations. However, unlike equations, with inequalities, we deal with ranges of values. Here’s a simple process to solve them:

• Manipulate the sides to isolate the variable term. This often includes adding or subtracting from both sides.
• Perform multiplication or division as needed. Be mindful of the operation's effect on the inequality.

For example, given the inequality \(-7x - 3 \leq 4\), we begin by addressing the constant term on the same side as the variable \(x\). Add 3 to both sides to simplify it to \(-7x \leq 7\). This operation allows us to get closer to isolating \(x\). It's crucial to follow each step carefully to ensure that the inequality maintains its balance and accurately reflects the range of solutions.

Reversing inequality signs

When working with inequalities, one crucial rule is that dividing or multiplying both sides by a negative number reverses the inequality sign. This rule is essential when solving inequalities because it ensures that the inequality remains true.
For instance, with the inequality \(-7x \leq 7\), to solve for \(x\), you divide both sides by -7. Doing so changes the inequality from \(\leq\) to \(\geq\). Thus, you end up with \(x \geq -1\).
Remember this rule as a fundamental part of solving inequalities. It ensures that your solution accurately represents the conditions set by the inequality. Failing to reverse the inequality sign when required can lead to incorrect solutions.

Isolating variables

Isolating the variable means getting the variable by itself on one side of the inequality or equation. This process helps in determining the value or range of values that satisfy the condition set by the inequality. Isolation usually involves the following steps:

• Clear constant terms from around the variable by performing the opposite arithmetic operation.
• If there's a coefficient next to the variable, divide or multiply to simplify. Always be alert to changes in inequality signs.

In the example \(-7x - 3 \leq 4\), to isolate \(x\), you first add 3 to each side, simplifying the inequality to \(-7x \leq 7\). This step helps us move closer to solving for \(x\). The process of isolation is critical because it lays the groundwork for finding the solution to the inequality.