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Chapter 1: Problem 51

Fill in the blank to form a correct inequality statement. If \(8 x>40,\) then \(x______5\)

Short Answer

Expert verified
x > 5

Step by step solution

01

- Understand the given inequality

The given inequality is: \(8x > 40\). This inequality shows that 8 times some value \(x\) is greater than 40.
02

- Isolate the variable \(x\)

To find the value of \(x\), divide both sides of the inequality by 8: \[ \frac{8x}{8} > \frac{40}{8} \]
03

- Simplify the inequality

Simplify the equation on both sides: \[ x > 5 \]

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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 an important skill in algebra. It's similar to solving equations, but instead of finding specific values, you determine a range of values. In our example, the inequality was \(8x > 40\). By finding which values of \(x\) make this inequality true, you're solving it. Think of inequality signs like balance scales. When you perform an action on one side, you must do the same on the other side to keep the inequality balanced.
Common actions include:
  • Adding or subtracting the same number to/from both sides
  • Multiplying or dividing both sides by the same number (be careful with negative numbers here!)
Solving inequalities opens up ways to understand constraints in real-world problems, like budgets or physical limits.
isolation of variables
To solve inequalities or equations, isolating the variable is key. To isolate a variable means to get it alone on one side of the inequality sign. In our problem:
\(8x > 40\)
We want to isolate \(x\). We do this by dividing both sides by 8:
\[ \frac{8x}{8} > \frac{40}{8} \]
Now, \(x\) is isolated:
\(x > 5\)
The variable \(x\) is now alone, and we have found that \(x\) must be greater than 5. It's a critical step because it allows us to find the range of values that makes the original inequality true.
simplifying inequalities
Simplifying inequalities makes them easier to understand and solve. Simplifying involves reducing the expression to its simplest form. In our problem:
\(8x > 40\)
We divided both sides by 8 to simplify:
\[ \frac{8x}{8} > \frac{40}{8} \]
This gives us the simplified inequality:
\(x > 5\)
The process of simplifying may involve combining like terms, factoring, or canceling common factors. It makes it clear which values of \(x\) satisfy the inequality. Always perform the same operation on both sides of the inequality sign to maintain the balance of the original statement.

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Most popular questions from this chapter

Challenge Problem Show that the real solutions of the equation \(a x^{2}+b x+c=0, a \neq 0,\) are the reciprocals of the real solutions of the equation \(c x^{2}+b x+a=0\), \(c \neq 0 .\) Assume that \(b^{2}-4 a c \geq 0\)

Fat Content Suppose that you order a small McCafeTM chocolate shake ( \(15 \mathrm{~g}\) of fat) and an Artisan Grilled Chicken Sandwich" (7 g of fat) at McDonald's. How many oatmeal raisin cookies can you eat ( \(5 \mathrm{~g}\) of fat) and still keep the total fat content of your meal to no more than \(47 \mathrm{~g} ?\)

Find the real solutions, if any, of each equation. Use any method. $$ x^{2}+x=4 $$

Find \(k\) so that the equation \(x^{2}-k x+4=0\) has a repeated real solution.

Find the real solutions, if any, of each equation. $$ 4(w-3)=w+3 $$
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