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Chapter 6: Problem 53

Solve the inequality. \(-\frac{a}{10} \leq-2\)

Short Answer

Expert verified
The solution to the inequality is \(a \geq 20\).

Step by step solution

01

Remove the negative sign by multiplying both sides by -1

Multiply both sides of the inequality by -1 while reversing the inequality sign. This gives \(\frac{a}{10} \geq 2\)
02

Clear the fraction

Multiply both sides by 10, to clear the fraction. This gives \(a \geq 20\).
03

Final Answer

The solution to the inequality is \(a \geq 20\), which means any number greater than or equal to 20 is a solution to the inequality.

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Key Concepts

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

Algebraic Expressions
Algebraic expressions form the backbone of much of algebra. They consist of numbers, variables, and arithmetic operations. An expression like \(-\frac{a}{10}\) identifies how the variable \(a\) interacts with a number (in this case, -1/10).
  • Variables: These are symbols, often letters like \(a\), that can represent different numbers in an expression.
  • Operations: The basic operations in algebraic expressions include addition, subtraction, multiplication, and division.
  • Constants: Numbers that remain the same, such as 10 in \(-\frac{a}{10}\), are constants.
Working with such expressions is fundamental in solving algebraic equations and inequalities, as you'll often need to rearrange and redefine different components to isolate variables. In this exercise, the task is to manipulate the given expression to find the range of values that satisfy the inequality.
Inequality Signs
Inequality signs are symbols used in mathematics to compare values. Knowing how they work is crucial when solving equations and inequalities.
  • "<" and ">": Represent 'less than' and 'greater than' respectively.
  • "\( \leq \) " and "\( \geq \)": These mean 'less than or equal to' and 'greater than or equal to', including the number itself in the solution set.
A key point when dealing with inequality signs is that they reverse when multiplying or dividing by a negative number. For instance, in this problem, the sign changes from less than or equal to greater than when we multiply both sides by -1. This is a crucial step that can easily be overlooked, but it is necessary to keep your solution correct.
Fractions in Equations
Fractions often appear in algebraic equations, and they require special attention. Solving an equation with fractions involves clearing the fractions to make the algebraic manipulation easier for further steps.
  • Multiplying to Clear Fractions: One way to eliminate fractions from an equation is by multiplying both sides by the denominator. In our problem, after changing the sign by multiplying by -1, multiplying by 10 helps clear the fraction.
  • Maintaining Equality: When altering an equation, it's crucial to perform the same operation to both sides to maintain balance. So, multiplying both sides of \(\frac{a}{10} \geq 2\) by 10 gives \(a \geq 20\).
These steps ensure that we can solve equations more straightforwardly, avoiding potential mistakes that come from dealing with fractions. It helps in finding the correct value or range of values for variables involved.

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Most popular questions from this chapter

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