Question

Solve the inequality. Write a sentence that describes the solution. $$ 8 \leq 2 x+6 \leq 18 $$

Short answer

The solution to the inequality is \(1 \leq x \leq 6\). This means \(x\) can be any number between 1 and 6, inclusive.

Step-by-step solution

Simplify the Inequality

First, subtract 6 from all parts of the inequality to simplify it. That results in \(2 \leq 2x \leq 12\).

Isolate the Variable

Next, divide all parts of the inequality by 2 to isolate \(x\). The result is \(1 \leq x \leq 6\).

Interpret the Results

This result means that \(x\) is a value that is greater than or equal to 1 and less than or equal to 6.

Key concepts

Inequality Notation

When solving inequalities, it's crucial to understand the notation used. Inequalities are mathematical expressions that describe the relationship between two values in terms of one being greater than, less than, or equal to the other. Symbols such as '<' for less than, '>' for greater than, '≤' for less than or equal to, and '≥' for greater than or equal to, are commonly used.
For instance, the inequality notation ‘8 ≤ 2x + 6 ≤ 18’ represents a range where the expression ‘2x + 6’ is at least 8 and at most 18. This type of notation is indicative of a compound inequality, which combines two inequalities into one statement. Understanding how to read this notation is the first step in being able to solve the inequality.
In practice, if you have an inequality like ‘a ≤ b ≤ c’, you'll deal with it as two separate inequalities: ‘a ≤ b’ and ‘b ≤ c’. Each part must be handled appropriately to maintain the balance of the inequality. This blend of literacy in symbols and operations is key to making sense of inequalities and moving towards finding a solution.

Isolate the Variable

To 'isolate the variable' means to get the variable on one side of the inequality sign, with everything else on the other side. This step is crucial for solving any inequality. The goal is to find the range of values that satisfy the inequality statement.
In our example, the compound inequality '8 ≤ 2x + 6 ≤ 18' requires us to isolate 'x'. To do this systematically, you perform the same operation across every part of the compound inequality. First, subtract 6 from all three parts, which simplifies the inequalities to '2 ≤ 2x ≤ 12'. Then, divide each part by 2, which isolates 'x', giving us '1 ≤ x ≤ 6'.

Remember the Rules

• If you multiply or divide by a negative number, you must flip the inequality signs.
• Perform identical mathematical operations on all sides of the inequality to maintain its truth.
• Keep track of each step for clarity and ease of verification.

These guidelines will ensure that your variable is properly isolated without disrupting the original inequality's truth.

Compound Inequalities

Compound inequalities involve two distinct inequalities that are connected either by the word 'and' or 'or'. In the 'and' case, such as with ‘8 ≤ 2x + 6 ≤ 18’, the solution is all values of 'x' that satisfy both inequalities simultaneously. This is often represented graphically on a number line, with a solid line covering all numbers between the two boundaries, inclusive, if the inequality uses '≤' or '≥'.

Breaking it Down

• Each part of a compound inequality can be solved separately, and then the intersection of the solutions provides the final answer.
• In our example, solving each inequality independently (after isolating 'x') would also give '1 ≤ x ≤ 6', overlapping sections of the number line that define the common solution set.
• In the case of 'or', the solution set includes values that satisfy at least one of the inequalities, which can lead to a union of solution sets.

Visual Aids

Using a number line for visualization is immensely helpful for understanding compound inequalities. It allows you to see the range of solution sets and determine the set of all possible values that satisfy the compound inequality.