Problem 92 - Algebra 1: Concepts and Skills

Question

Solve the inequality. Then graph the solution. (Lesson 6.2) $$6 x \leq-2$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x \leq -1/3)

Step 1: Isolate the Variable 'x'

To isolate 'x', divide both sides of the inequality by 6. This will give us \(x \leq -2/6\)

Step 2: Simplify the Right Side of the Inequality

Simplify the right side of the inequality by expressing -2/6 in simplest form. Dividing -2 by 6 yields \(x \leq -1/3\)

Step 3: Graph the Solution

Draw a number line, and place a dot at -1/3, with the dark area of the number line extending to the left of -1/3 to indicate that 'x' is less than or equal to -1/3.

Key Concepts

Solving Inequalities

Solving inequalities is a fundamental concept in algebra that involves finding the range of possible values for a variable that make an inequality true. The process of solving an inequality is very similar to solving an equation, but with an important distinction: if we multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes.
For example, in the given inequality \(6x \leq -2\), we start by isolating the variable. This means we need to get \(x\) by itself on one side of the inequality. To do this, divide both sides by 6:

• \(\frac{6x}{6} \leq \frac{-2}{6}\)
• This simplifies to \(x \leq -\frac{1}{3}\)

Remember not to flip the inequality sign here, because we're dividing by a positive number. The solution \(x \leq -\frac{1}{3}\) tells us that \(x\) can be any number that is less than or equal to \(-\frac{1}{3}\). This completes the process of solving the inequality.

Graphing Solutions

Graphing the solutions of an inequality helps us visualize the range of values that satisfy the inequality. Once you have solved the inequality algebraically, like reaching the solution \(x \leq -\frac{1}{3}\), you will want to represent this on a number line.
To graph \(x \leq -\frac{1}{3}\):

• Draw a horizontal line to represent your number line.
• Identify the point \(-\frac{1}{3}\) on this line.
• Since \(x\) can be equal to \(-\frac{1}{3}\), place a solid dot on \(-\frac{1}{3}\).
• To indicate that \(x\) can also be less than \(-\frac{1}{3}\), shade the line to the left of \(-\frac{1}{3}\).

This graph shows that any number on the shaded part of the number line is a solution to the inequality \(6x \leq -2\). Graphing is helpful as it gives us clear visual confirmation of the possible solutions.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operations (such as addition, subtraction, multiplication, and division). In the inequality \(6x \leq -2\), \(6x\) is an algebraic expression representing six times a variable \(x\).
When working with inequalities, you are often asked to manipulate algebraic expressions to isolate the variable. This involves combining like terms, performing arithmetic operations on both sides, and using properties of equality and inequalities to isolate variables. Here’s how it relates to our original problem:

• The expression \(6x\) indicates the multiplication of \(x\) by 6.
• To solve the inequality, we treated this expression similar to an equation by dividing it by 6, simplifying the inequality to \(x \leq -\frac{1}{3}\).

Understanding how to manage algebraic expressions is crucial for solving and simplifying inequalities. Strong skills in handling these ensure efficient and correct solutions in algebra.