Problem 90 - Algebra 1: Concepts and Skills

Question

Solve the inequality. \(2<2 x+7\)

Step-by-Step Solution

Answer

The solution to the inequality is \(x > -2.5\).

Step 1: Isolate the term with an \(x\)

To do this, subtract 7 from both sides of the inequality to isolate the term with \(x\).\n \(2 - 7 < 2x + 7 - 7 \)\nThis simplifies to\n \(-5 < 2x\).

Step 2: Isolate \(x\)

Now, divide both sides of the inequality by 2 to isolate \(x\).\n \(-5/2 < x\).\n In other words, \(x > -5/2\) or \(x > -2.5\).

Key Concepts

Isolation of Variables

When solving inequalities, one of the primary goals is to isolate the variable in question. In our example exercise, we are tasked with solving the inequality \(2 < 2x + 7\). To begin isolating the variable \(x\), we first need to eliminate any other numbers from the side of the inequality where \(x\) lives. This process involves reversing the operations that have been applied to \(x\).

• Start by identifying terms that aren't attached to \(x\). Here, the term is \(+7\).
• Subtract 7 from both sides to preserve the balance of the inequality. Thus, the equation \(2 - 7 < 2x + 7 - 7\) becomes \(-5 < 2x\).
• Next, to fully isolate \(x\), divide each side by the coefficient of \(x\), which is 2 in this case. You end up with \(-\frac{5}{2} < x\).

By following these steps, we've successfully isolated \(x\) on one side of the inequality, making it easier to interpret and apply.

Linear Inequalities

Linear inequalities, like the one presented in our exercise, express a relationship where one side of the inequality can be larger or smaller than the other. These inequalities are pivotal in algebra due to their simplicity and frequency in application.

• They take the general form \(ax + b < c\), and can similarly use \(>\), \(\leq\), or \(\geq\).
• Solving a linear inequality usually involves the same steps as solving a linear equation: simplifying, isolating the variable, and ensuring the inequality is maintained.
• The main difference from equations is the treatment of inequalities, particularly when multiplying or dividing by negative numbers (which can reverse the inequality sign).

In the given example, after isolating the variable \(x\), we deduced that the inequality holds when \(x\) is greater than \(-2.5\), characterizing a segment of all possible values \(x\) can take, rather than a singular solution.

Inequality Properties

Understanding the properties of inequalities is crucial for their effective manipulation and solving. These properties allow us to transform and simplify inequalities while maintaining their intrinsic truths.

• Transitive Property: If \(a < b\) and \(b < c\), then \(a < c\). It helps connect relationships between multiple inequalities.
• Addition/Subtraction Property: You can add or subtract the same value from both sides of an inequality without changing its direction. This property was used when subtracting 7 in the original exercise.
• Multiplication/Division Property: Multiplying or dividing both sides of an inequality by a positive number retains the direction of the inequality. However, if the number is negative, the inequality sign reverses. In this exercise, we divided by a positive number, so the inequality direction stayed intact.

By applying these properties diligently, we ensure the solutions to inequalities are correct and logical, as demonstrated by deducing that \(x > -2.5\). This understanding helps in graphing, interpreting, and applying inequalities across various mathematical problems.