Problem 84 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each inequality. $$x+4

Step-by-Step Solution

Answer

The inequality \(x+4

Step 1: Simplify the inequality

The first step is to simplify the inequality. To do this, an amount, equal to the coefficient before x on the left-hand side, is subtracted. This is 4 in this case. So, subtract 4 from both sides \(x+4-4

Step 2: Solve

Then, the inequality simplifies to \(x < x + 6\). Now, we subtract x from both sides to solve for x, \(x - x < x - x + 6\). This simplifies to \(0 < 6\). This inequality is always true for any real number.

Step 3: Final Answer

Since the resulting inequality doesn't contain x, this means that the original inequality is true for any real x.

Key Concepts

Inequality Simplification

When working with inequalities, simplification is a crucial step towards finding the solution. Inequality simplification involves reducing the inequality to its simplest form to make it easier to solve. You may need to combine like terms, which are terms that have the same variable raised to the same power, or you might need to eliminate terms that appear on both sides of the inequality.
In the given exercise, the initial inequality is \(x+4

Algebraic Inequalities

Algebraic inequalities are equations that use inequality symbols (greater than >, less than <, greater than or equal to ≥, and less than or equal to ≤) instead of an equals sign. They show a relationship between two expressions and indicate that one expression is larger or smaller than the other. The key to solving algebraic inequalities lies in understanding that the inequality sign needs to stay valid throughout all manipulation steps.

• Always perform the same operation on both sides of the inequality to maintain the balance.
• When multiplying or dividing by a negative number, reverse the direction of the inequality.
• Remember the possible range of solutions rather than a single solution, as often inequalities have multiple solutions.

For our exercise, since we end with the statement \(0<6\), which does not include the variable \(x\), it indicates that this inequality holds true for any real value of \(x\), which is why it's an example of an identity inequality.

Steps in Solving Inequalities

Solving inequalities typically follows a structured approach, and being familiar with this approach is vital for tackling algebraic problems successfully. The steps often include:

1. Isolate the variable: Use addition or subtraction to get the variable on one side of the inequality.
2. Simplify: Combine like terms and simplify both sides of the inequality as much as possible.
3. Divide or multiply: If the variable has a coefficient, divide or multiply both sides of the inequality to solve for the variable. Remember to flip the inequality sign if multiplying or dividing by a negative number.
4. Check your solution: Plug the found solution back into the original inequality to ensure it makes the inequality true.

In the provided example, the steps taken led us to the simplified form, which after subtracting \(x\) from both sides, resulted in \(0<6\), a statement that is always true. This conclusion provides us with important information: our original inequality \(x+4

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Problem 83

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Problem 84

Other exercises in this chapter

Problem 83

Use the given information to write an equation. Let \(x\) represent the number described in each exercise. Then solve the equation and find the number. When one

View solution

Problem 83

In Exercises \(79-82,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

View solution

Problem 84

When the measure of a given angle is added to three times the measure of its complement, the sum equals the sum of the measures of the complement and supplement

View solution

Problem 84

Write as an algebraic expression in which \(x\) represents the number: the quotient of 9 and a number, decreased by 4 times the number. (Section 1.1, Example 3)

View solution

Key Concepts

Inequality Simplification

When working with inequalities, simplification is a crucial step towards finding the solution. Inequality simplification involves reducing the inequality to its simplest form to make it easier to solve. You may need to combine like terms, which are terms that have the same variable raised to the same power, or you might need to eliminate terms that appear on both sides of the inequality.
In the given exercise, the initial inequality is \(x+4

Algebraic Inequalities

Algebraic inequalities are equations that use inequality symbols (greater than >, less than <, greater than or equal to ≥, and less than or equal to ≤) instead of an equals sign. They show a relationship between two expressions and indicate that one expression is larger or smaller than the other. The key to solving algebraic inequalities lies in understanding that the inequality sign needs to stay valid throughout all manipulation steps.

• Always perform the same operation on both sides of the inequality to maintain the balance.
• When multiplying or dividing by a negative number, reverse the direction of the inequality.
• Remember the possible range of solutions rather than a single solution, as often inequalities have multiple solutions.

For our exercise, since we end with the statement \(0<6\), which does not include the variable \(x\), it indicates that this inequality holds true for any real value of \(x\), which is why it's an example of an identity inequality.

Steps in Solving Inequalities

Solving inequalities typically follows a structured approach, and being familiar with this approach is vital for tackling algebraic problems successfully. The steps often include:

1. Isolate the variable: Use addition or subtraction to get the variable on one side of the inequality.
2. Simplify: Combine like terms and simplify both sides of the inequality as much as possible.
3. Divide or multiply: If the variable has a coefficient, divide or multiply both sides of the inequality to solve for the variable. Remember to flip the inequality sign if multiplying or dividing by a negative number.
4. Check your solution: Plug the found solution back into the original inequality to ensure it makes the inequality true.

In the provided example, the steps taken led us to the simplified form, which after subtracting \(x\) from both sides, resulted in \(0<6\), a statement that is always true. This conclusion provides us with important information: our original inequality \(x+4

Previous

Problem 83

Next

Problem 84

Other exercises in this chapter

Problem 83

Use the given information to write an equation. Let \(x\) represent the number described in each exercise. Then solve the equation and find the number. When one

View solution

Problem 83

In Exercises \(79-82,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

View solution

Problem 84

When the measure of a given angle is added to three times the measure of its complement, the sum equals the sum of the measures of the complement and supplement

View solution

Problem 84

Write as an algebraic expression in which \(x\) represents the number: the quotient of 9 and a number, decreased by 4 times the number. (Section 1.1, Example 3)

View solution