Problem 81 - Elementary Algebra

Question

What integers can replace \(x\) so that the statement \(-6

Step-by-Step Solution

Answer

Question: List down the integer values of x that satisfy the inequality \(-6 < x < -2\). Answer: The integer values that satisfy the inequality are -5, -4, and -3.

Step 1: Understanding the inequality

In this statement, \(-6

Step 2: Listing the integer values

Since x needs to be greater than -6 and less than -2, let's list down the integer values within that boundary. -5, -4, and -3 These integers are the values that satisfy the inequality \(-6 < x < -2\).

Key Concepts

Integer Values

When we talk about integer values, we're looking at whole numbers that can be positive, negative, or zero, but without any fractions or decimals. An important aspect of solving inequalities is to identify which integers satisfy the given conditions.

In the exercise provided, the challenge is to find the integers that are greater than -6 and simultaneously less than -2. It's pretty much like looking for the numbers that are sitting between -6 and -2 on the number line. The integers that lie snugly in this range without touching the boundaries are -5, -4, and -3. These are the only integers that make the statement \( -6 < x < -2 \) true. It is crucial to leave out the endpoints, -6 and -2, because the inequality is strict—with the < signs indicating 'strictly greater than' and 'strictly less than'.

Solving Inequalities

The process of solving inequalities might seem daunting at first, but it's similar to solving equations, with a key difference in how we treat the inequality when multiplying or dividing by a negative number. When confronted with an inequality, such as the one in our exercise \( -6 < x < -2 \) we're looking for a set of numbers that make the inequality true, rather than a single solution like in an equation.

It's helpful to visualize the number line and shade in the region that satisfies the inequality. Remember to use open circles to indicate that the end values aren't included if the inequality doesn't have an 'equals to' part, like \( \leq \) or \( \geq \) symbols. In this case, integers are distinct points on the number line and picking the correct ones within the given range is critical. Be sure to consider the direction of the inequality, as flipping an inequality sign can also flip your solution set.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operators. For example, \( 2x - 5 \) is an algebraic expression with \( x \) as the variable. When we're solving inequalities that involve algebraic expressions, we apply similar operations as we do with arithmetic expressions—but we keep an eye on the direction of the inequality sign.

If the inequality involves a simple algebraic expression without the need to conduct operations on both sides, such as the expression given in our exercise \( -6 < x < -2 \) the solution involves understanding the relationship presented by the inequality and determining which values for the variable \( x \) maintain the truth of the entire expression. Working with algebraic expressions requires careful manipulation; one must be mindful of each step, especially when distributing or combining like terms.

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Problem 81

Other exercises in this chapter

Problem 80

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ \frac{4}{x^{-6}} $$

View solution

Problem 80

A person borrows $$\$ 11.00$$ on Monday and then pays back $$\$ 8.00$$ on Tuesday. How much does this person owe?

View solution

Problem 81

Find the value of each of the following expressions. \(m=\frac{2 s+1}{T} . \quad\) Find \(m\) if \(s=-8\) and \(T=5\)

View solution

Problem 81

Find the product for the following problems. Write the result in scientific notation. $$ \left(3 \times 10^{6}\right)\left(7 \times 10^{7}\right) $$

View solution

Key Concepts

Integer Values

When we talk about integer values, we're looking at whole numbers that can be positive, negative, or zero, but without any fractions or decimals. An important aspect of solving inequalities is to identify which integers satisfy the given conditions.

In the exercise provided, the challenge is to find the integers that are greater than -6 and simultaneously less than -2. It's pretty much like looking for the numbers that are sitting between -6 and -2 on the number line. The integers that lie snugly in this range without touching the boundaries are -5, -4, and -3. These are the only integers that make the statement \( -6 < x < -2 \) true. It is crucial to leave out the endpoints, -6 and -2, because the inequality is strict—with the < signs indicating 'strictly greater than' and 'strictly less than'.

Solving Inequalities

The process of solving inequalities might seem daunting at first, but it's similar to solving equations, with a key difference in how we treat the inequality when multiplying or dividing by a negative number. When confronted with an inequality, such as the one in our exercise \( -6 < x < -2 \) we're looking for a set of numbers that make the inequality true, rather than a single solution like in an equation.

It's helpful to visualize the number line and shade in the region that satisfies the inequality. Remember to use open circles to indicate that the end values aren't included if the inequality doesn't have an 'equals to' part, like \( \leq \) or \( \geq \) symbols. In this case, integers are distinct points on the number line and picking the correct ones within the given range is critical. Be sure to consider the direction of the inequality, as flipping an inequality sign can also flip your solution set.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operators. For example, \( 2x - 5 \) is an algebraic expression with \( x \) as the variable. When we're solving inequalities that involve algebraic expressions, we apply similar operations as we do with arithmetic expressions—but we keep an eye on the direction of the inequality sign.

If the inequality involves a simple algebraic expression without the need to conduct operations on both sides, such as the expression given in our exercise \( -6 < x < -2 \) the solution involves understanding the relationship presented by the inequality and determining which values for the variable \( x \) maintain the truth of the entire expression. Working with algebraic expressions requires careful manipulation; one must be mindful of each step, especially when distributing or combining like terms.