Problem 20 - Prealgebra

Question

For the given value, state whether each inequality is true or false. $$5 k>35, k=7$$

Step-by-Step Solution

Answer

The inequality is false.

Step 1: Substitute the Given Value

First, we need to replace the variable \( k \) in the inequality with the given value. The problem states \( k = 7 \), so we substitute 7 into the inequality \( 5k > 35 \). This gives us the expression: \( 5(7) > 35 \).

Step 2: Simplify the Expression

Now simplify the expression \( 5(7) \). Multiply 5 by 7 to get 35. Now the inequality reads \( 35 > 35 \).

Step 3: Analyze the Inequality

Now we need to determine if the inequality \( 35 > 35 \) is true or false. Since 35 is not greater than 35, the inequality is false.

Key Concepts

Understanding Inequalities

Inequalities are like equations, but instead of an equal sign, they use symbols like \(<, >, \leq,\) and \(\geq\) to compare values. They tell us whether one value is smaller, greater, or not exactly equal to another value.
For example, when you see \(5k > 35\), it means "5 times \(k\) should be greater than 35." It’s important to note that "greater than" and "less than" do not include equality, while "greater than or equal to" does.

• "Greater than" (\(>\)) means the first number is larger.
• "Greater than or equal to" (\(\geq\)) means the first number is either larger or exactly equal.

In the exercise, we compared \(5k\) with 35 using a \(>\) sign, expecting \(5k\) to be strictly more than 35. Knowing these symbols is crucial for correctly interpreting and solving inequality problems.

The Substitution Method

The substitution method is a helpful way to solve equations and inequalities by replacing variables with known values. This technique helps in simplifying the expression you're dealing with.
In our given exercise, we had the inequality \(5k > 35\) and knew that \(k = 7\). Substitution involves just putting the number 7 in place of \(k\), transforming our inequality into \(5(7) > 35\).

• Identify the variable: Look for \(k\) in the inequality.
• Substitute the value: Replace \(k\) with 7.
• Reevaluate the expression to see if it makes the inequality true or false.

This method makes the comparison process straightforward because you're dealing strictly with numbers, removing any variables.

Simplifying Expressions

Simplifying expressions makes them easier to handle and solve. This process involves performing operations to bring equations or inequalities to their simplest form.
When we simplified \(5(7) > 35\), the multiplication \(5 \times 7\) was carried out to result in 35. This left us with the simpler statement: does \(35 > 35\)?

• Identify the operations: In this case, multiplication was involved.
• Perform the calculations: Calculate \(5 \times 7 = 35\).
• Answer the question of equality: Evaluate whether the final statement is true or false.

In our example, the final expression simplifies to \(35 > 35\), which is false.
Simplification is critical as it reveals the true nature of the relationship between the numbers involved and helps in clearly determining the truth of the inequality.