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Chapter 3: Problem 41

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ 4 x-3 \leq 21 $$

Short Answer

Expert verified
\( x \leq 6 \)

Step by step solution

01

Isolate the variable term

Begin by isolating the term with the variable on one side of the inequality. In this case, we need the term '4x' to be by itself on the left side. To do this, add 3 to both sides:\[ 4x - 3 + 3 \leq 21 + 3 \]This simplifies to:\[ 4x \leq 24 \]
02

Solve for the variable

Now, solve for \(x\) by dividing both sides of the inequality by 4, the coefficient of \(x\):\[ \frac{4x}{4} \leq \frac{24}{4} \]This simplifies to:\[ x \leq 6 \]

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Key Concepts

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

Solving Inequalities
Solving inequalities is a fundamental skill in algebra that helps us find all the possible values of a variable that satisfy a mathematical statement. Unlike equations, which have an equal sign, inequalities use signs like ">", "<", "鈮�", and "鈮�" to show the relationship between expressions. When you solve an inequality, you are trying to find the set of values that make the inequality true.

Here's a simple breakdown of solving an inequality:
  • First, treat the inequality much like an equation. Think about how you would isolate the variable if it were an equation.
  • Remember that multiplying or dividing by a negative number reverses the sign of the inequality, which is a crucial difference from solving regular equations.
  • Consider all possible values for the variable, since inequalities often mean the variable could have several acceptable values, unlike equations that usually have a finite set of solutions.
Understanding these rules is essential, as they guide your steps in solving and interpreting inequalities correctly.
Isolation of Variable
In solving inequalities, isolating the variable means rearranging the inequality so that the variable appears alone on one side of the inequality sign. This is crucial because it helps us see directly what values the variable might take.

Here's a step-by-step guide:
  • Identify the term with the variable on one side. In our example, it's the term "4x" on the left-hand side.
  • Use basic arithmetic operations to move other terms over to the opposite side. For instance, in the inequality 4x - 3 鈮� 21, we add 3 to both sides to eliminate the "-3" next to the variable.
  • Keep the inequality balanced by performing the same operation on both sides, just like you would with an equation.
By following these steps, you ensure the inequality remains true as you simplify it to find the solution set.
Algebraic Manipulation
Algebraic manipulation involves using mathematical operations and properties to change the form of an expression or inequality to make it easier to work with. This concept is key in solving inequalities as it allows us to isolate and solve for the variable.

Consider these manipulation techniques:
  • Combine like terms and simplify your expressions where possible to make equations less complex.
  • Use inverse operations to cancel out terms. For instance, add to cancel out subtractions, and multiply to cancel out divisions.
  • Factor expressions if needed, although factoring is less common with simple linear inequalities.
  • Always be careful with division or multiplication by negative numbers, as this reverses the inequality sign.
By mastering these techniques, you can skillfully solve a range of algebraic inequalities, from the very simple to the more complex.

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Most popular questions from this chapter

Solve each equation. $$7(x+2)=21$$

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Solve each equation. $$5(2 x-1)-(3 x+4)=4(x+3)-27$$

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