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Chapter 5: Problem 66

Write an inequality to represent the statement. \(y\) is no more than 7 .

Short Answer

Expert verified
y \leq 7

Step by step solution

01

Identify the key phrase

Recognize that the phrase 'no more than' indicates that we are dealing with an inequality where one side is less than or equal to the other side.
02

Determine the inequality symbol

The phrase 'no more than' means that the value on the left side of the inequality can be less than or equal to the value on the right side. Therefore, the inequality symbol used is \( \leq \).
03

Translate the statement into an inequality

Replace the phrase 'no more than' with the inequality symbol \( \leq \) to form the inequality: \( y \leq 7 \).

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Key Concepts

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

Inequality Symbols
In algebra, inequality symbols are used to compare the sizes of two values. These symbols are just as important as the equal sign. Inequality symbols include:
  • \( < \): less than
  • \( > \): greater than
  • \( \leq \): less than or equal to
  • \( \geq \): greater than or equal to
These symbols allow us to express a range of possible values for a variable. For instance, if you know that a value is no more than 7, you would use the symbol \( \leq \) to show that the value can be 7 or any number less than 7.
Translating Phrases into Inequalities
When working with word problems, it鈥檚 important to translate phrases into mathematical language. In our example, the phrase 'no more than' indicates that the quantity we're talking about is less than or equal to a certain value. Here's a general guideline for translation:
  • 'less than' translates to \( < \)
  • 'greater than' translates to \( > \)
  • 'no more than' or 'at most' translates to \( \leq \)
  • 'at least' translates to \( \geq \)
Understanding these phrases will help you accurately translate verbal expressions into mathematical inequalities.
Solving Inequalities
Solving inequalities is similar to solving equations, but there are a few differences to keep in mind. When you add or subtract the same number from both sides of an inequality, the inequality remains unchanged. However, when you multiply or divide both sides by a negative number, you must reverse the inequality symbol. Here's an example with the inequality \( y \leq 7 \):
  • Subtract 3 from both sides: \( y - 3 \leq 7 - 3 \Rightarrow y - 3 \leq 4 \)
  • Divide both sides by \( -1 \) (note the inequality reversal): \( -y - 3 \geq -4 \)
  • Simplify: \( y \leq 4 \)
These steps show how to manipulate inequalities while keeping the inequalities true.
Algebraic Expressions
In algebra, expressions are combinations of variables, numbers, and operations. They don鈥檛 include equality or inequality signs. For example, \(2x + 3\) is an algebraic expression. When dealing with inequalities, these expressions can be on one or both sides of the inequality symbol. In our problem, the algebraic expression is simply \(y\), which must be less than or equal to 7. These concepts form the foundation for writing and solving inequalities, as inequalities often compare an algebraic expression to a number.

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Most popular questions from this chapter

Solve the system of equations by using the addition method. (See Examples \(3-4)\) $$ \begin{array}{l} 2 x+11 y=4 \\ 3 x-6 y=5 \end{array} $$

Solve the system. $$ \begin{array}{l} (x-1)^{2}+(y+1)^{2}=5 \\ x^{2}+(y+4)^{2}=29 \end{array} $$

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{aligned} 2 x-y &=8 \\ x-\frac{1}{2} y &=4 \end{aligned} $$

Solve the system of equations by using the substitution method. (See Example 2\()\) $$ \begin{array}{rr} x+3 y= & 5 \\ 3 x-2 y= & -18 \end{array} $$

A radiator has \(16 \mathrm{~L}\) of a \(36 \%\) antifreeze solution. How much must be drained and replaced by pure antifreeze to bring the concentration level up to \(50 \%\) ?
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