Problem 64 - Algebra 2

Question

Solve each inequality. $$ 5 < 2 x+7 < 13 $$

Step-by-Step Solution

Answer

The solution is \(-1 < x < 3\).

Step 1: Break Down the Compound Inequality

To solve the compound inequality \(5 < 2x + 7 < 13\), we need to break it down into two separate inequalities: \(5 < 2x + 7\) and \(2x + 7 < 13\). This will allow us to solve each part individually.

Step 2: Solve the First Inequality

Start with the first inequality: \(5 < 2x + 7\). To isolate \(2x\), subtract \(7\) from both sides: \(5 - 7 < 2x\) simplifying to \(-2 < 2x\). Divide both sides by \(2\) to solve for \(x\): \(-1 < x\).

Step 3: Solve the Second Inequality

Now solve the second inequality: \(2x + 7 < 13\). Subtract \(7\) from both sides to isolate \(2x\): \(2x < 13 - 7\), which simplifies to \(2x < 6\). Divide each side by \(2\) to find \(x\): \(x < 3\).

Step 4: Combine the Inequalities

Combine both results from Step 2 and Step 3: \(-1 < x < 3\). This shows the solution set where \(x\) is greater than \(-1\) and less than \(3\).

Step 5: Interpret the Results

The solution \(-1 < x < 3\) indicates that \(x\) is any real number between \(-1\) and \(3\), but not including \(-1\) or \(3\).

Key Concepts

Compound Inequality

A compound inequality is a mathematical expression that involves two separate inequalities joined together. In the exercise we have, the compound inequality is given as \(5 < 2x + 7 < 13\). This compound expression essentially consists of two individual inequalities:

• \(5 < 2x + 7\)
• \(2x + 7 < 13\)

Compound inequalities are used to express a range of values that satisfy both inequalities simultaneously. In this case, we're seeking the values of \(x\) that fit both the conditions above.
To solve them, we'll need to break them down into the separate inequalities and solve each individually, which is also the first step of our approach.
Once you have your individual solutions, you can then combine them to form the full solution to the original compound inequality.

Algebra

Algebra is the branch of mathematics that deals with expressions and equations, including the use of variables like \(x\) in our compound inequality problem. Solving inequalities is a key algebraic concept, which involves manipulating these expressions to isolate variables and find their possible values.
The process often involves:

• Using operations like addition, subtraction, multiplication, or division.
• Following specific rules to preserve the inequality (for example, flipping the inequality sign when multiplying or dividing by a negative number).

In the exercise, we solve each part of the compound inequality by isolating \(x\). This requires subtraction to remove constants and division to solve for the variable.
Algebraic principles guide us in rearranging terms and maintaining equality/inequality while simplifying the expressions step by step.

Solution Set

The solution set represents all possible values that satisfy an inequality or equation. It's a crucial concept when solving inequalities like \(5 < 2x + 7 < 13\).
By solving the two separate inequalities, \(5 < 2x + 7\) and \(2x + 7 < 13\), we found two results:

• \(-1 < x\)
• \(x < 3\)

The overlapping region of these results forms the solution set \(-1 < x < 3\). It means that \(x\) can be any number between \(-1\) and \(3\), not including the endpoints - essentially all real numbers in that range.
Understanding the solution set is essential because it gives the range of values that fulfills the entire compound inequality, illustrating both parts of the expression in one comprehensive answer.