Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Copy and complete the statement using the correct inequality symbol. If \(2-4 x>-10\), then \(x\)______3.

Step-by-Step Solution

Answer

If \(2-4x > -10\), then \(x < 3\).

Step 1: Isolate 'x'

To isolate 'x', first bring '4x' to the right side of the inequality, which leaves the equation as follows: \(2 > 4x - 10\). Then add '10' to both sides to get \(12 > 4x\). Finally, divide both sides by '4' to isolate 'x'. This leaves us with \(x < 3\).

Step 2: Comparing 'x' with '3'

The inequality symbol between 'x' and '3' is '<', indicating 'x' should be less than '3'.

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols, often represented by letters, to solve equations and find unknown values. These letters, known as variables, stand for numbers and allow us to generalize mathematical problems.

When working with algebra, typically the goal is to solve equations for a specific variable. You'll encounter statements that include both numbers and variables, and by applying algebraic principles, you can manipulate these statements to find solutions.

• Variables: Symbols representing unknown numbers
• Equations: Mathematical statements that assert two expressions are equal
• Operations: Include addition, subtraction, multiplication, and division

With algebra, equations can become more complex, involving multiple steps to isolate variables and reveal their values. Mastering these skills provides a foundation for solving many types of mathematical problems.

Solving Inequalities

Solving inequalities is similar to solving equations, but instead of having an equal sign, inequalities use symbols such as '>', '<', '≥', and '≤'. These symbols express that one side is greater than, less than, or specifically not equal to the other side.

The main difference when solving inequalities is that, when multiplying or dividing by a negative number, you must flip the inequality sign. This adjustment keeps the inequality truthful. Working through inequalities involves:

• Isolating the variable: Move terms around to get the variable alone on one side
• Applying algebraic operations: Use addition, subtraction, multiplication, or division
• Reversing the sign: If dividing or multiplying by negative numbers

Through these steps, you'll find the range of values that satisfy the inequality rather than a single solution as in regular equations. Inequalities often require careful attention to detail to ensure the solution is correct, reflecting the nature of relationships rather than equalities.

Linear Equations

Linear equations are an important concept in algebra. They are equations that form a straight line when graphed and can be written in the form of 'Ax + B = C', where 'A', 'B', and 'C' are constants, and 'x' is the variable.

Solving linear equations involves finding the value of 'x' that makes the equation true. The process is relatively straightforward and involves several steps:

• Identify the equation: Write down the expression you need to solve
• Isolate the variable: Use addition, subtraction, multiplication, or division to get 'x' alone on one side of the equation
• Check your work: Substitute your solution back into the original equation to ensure it balances

Linear equations are simple yet fundamental, forming the basis for understanding more complex mathematical concepts. They provide clear insights into the relationships between different quantities and are widely applicable in various real-world situations.