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Chapter 6: Problem 9

Solve the inequality. $$5 x+12 \leq 62$$

Short Answer

Expert verified
The solution to the inequality is \(x \leq 10\).

Step by step solution

01

Subtract 12 from both sides

Subtract 12 from each side of the inequality in order to isolate the term with x. This results in the new inequality \(5x \leq 50\)
02

Divide by 5

Now, divide each side of the inequality by 5 to solve for x. This gives us the solution \(x \leq 10\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Algebra
Algebra is all about working with numbers and variables to find unknown values. Unlike basic arithmetic, which uses specific numbers, algebra uses letters to represent these unknowns. For example, in the inequality \(5x + 12 \leq 62\), \(x\) represents a variable. This allows us to find solutions that satisfy the inequality.

In algebra, you perform operations like addition, subtraction, multiplication, and division to simplify expressions and solve equations or inequalities. Here are some key skills to master in algebra:
  • Understanding and manipulating expressions
  • Solving equations and inequalities
  • Using variables to represent numbers
Algebra is a fundamental tool in mathematics, useful for everything from simple problem solving to complex analysis.
Exploring Linear Inequalities
Linear inequalities involve mathematical statements that use inequality symbols (like \(\leq\), \(<\), \(\geq\), and \(>\)) instead of an equal sign. These symbols tell us about the relationship between expressions. In the exercise \(5x + 12 \leq 62\), we're dealing with a linear inequality because neither expression is squared or cubed.

To solve linear inequalities:
  • Use similar steps as solving equations: add, subtract, multiply, or divide.
  • The goal is to isolate the variable, making it clear what values satisfy the inequality.
  • Be cautious when multiplying or dividing by negative numbers, as this reverses the inequality's direction.
For example, solving \(5x \leq 50\) means finding all \(x\) values that make the inequality true, like \(x = 10\) or any number smaller.
Isolating Variables in Inequalities
Isolating variables means getting the variable (usually \(x\)) by itself on one side of the inequality, making it easier to see which values satisfy the inequality.

For the inequality \(5x + 12 \leq 62\), start by subtracting 12 from both sides to make the inequality simpler:
\[5x \leq 50\]
Next, divide each side by 5 to isolate \(x\):
\[x \leq 10\]Now, the solution is clear: any value of \(x\) less than or equal to 10 satisfies the original inequality.

When isolating variables:
  • Perform the same operation on both sides to keep the inequality balanced.
  • Keep track of operations carefully to avoid mistakes.
  • Reassess your solution to make sure it fits the original problem.
Understanding how to isolate variables helps in solving not just inequalities, but a wide range of algebraic problems.

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