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Chapter 2: Problem 13

Determine whether inequality is true or false. \(9-(-3) \leq 12\)

Short Answer

Expert verified
The inequality is true.

Step by step solution

01

Simplify the expression on the left side

Start by simplifying the expression on the left side of the inequality. We have: \( 9 - (-3) \). Remember that subtracting a negative number is equivalent to adding its positive counterpart.
02

Perform the addition

Next, perform the addition: \( 9 + 3 \) which simplifies to \( 12 \). So, the inequality now reads: \( 12 \leq 12 \).
03

Evaluate the inequality

Compare the simplified left side to the right side of the inequality. Since both sides are equal, the inequality \( 12 \leq 12 \) is true.

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Key Concepts

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

Simplifying Expressions
The first step in solving any mathematical inequality is to simplify the expressions on both sides. In our exercise, we start with the expression \( 9 - (-3) \). Simplifying expressions means breaking them down into their simplest form.
Here, subtracting a negative number can be tricky. Fortunately, a good rule to remember is: subtracting a negative is like adding a positive. So, \( 9 - (-3) \) becomes \( 9 + 3 \).
After simplification, we get \( 9 + 3 = 12 \). This makes our inequality much easier to handle. Simplified expressions are less prone to errors and make subsequent steps more manageable.
Inequality Evaluation
Once we have simplified the expressions, the next step is to evaluate the inequality. After simplifying, our inequality reads \( 12 \leq 12 \).
To assess if this is true, we determine if the left side (LS) is really less than or equal to the right side (RS). Here are simple rules:
  • If LS is less than RS, the inequality is true.
  • If LS is equal to RS, the inequality is also true.
  • If LS is greater than RS, the inequality is false.
In our case, both sides are equal (12 = 12), therefore the inequality \( 12 \leq 12 \) is indeed true.
Evaluating inequalities accurately ensures that you draw the correct conclusion every time.
Negative Numbers
Dealing with negative numbers often confuses students, but understanding them is crucial in many math problems. In our exercise, we encounter \( -(-3) \). Here鈥檚 a quick guide to handle negative numbers:
  • Negative signs indicate a direction opposite to positive numbers on a number line.
  • Subtracting a negative number means moving in the positive direction.
  • Thus, \( -(-3) \) becomes positive 3.
By keeping these points in mind, you'll find it easier to handle negative numbers in any context.
Also remember, when you multiply or divide by a negative number in inequalities, the inequality sign flips. Although not needed here, being aware of this rule is helpful for more complex problems.

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Most popular questions from this chapter

Show a complete solution to each problem. Increasing the percentage. A mechanic has found that a car with a 16 -quart radiator has a \(40 \%\) antifreeze mixture in the radiator. She has on hand a \(70 \%\) antifreeze solution. How much of the \(40 \%\) solution would she have to replace with the \(70 \%\) solution to get the solution in the radiator up to \(50 \% ?\)

Solve each problem. If the circumference of a circle is \(3 \pi\) meters, then what is the radius?

Solve each problem by using a compound inequality. See Example 12. Supply and demand. The function \(S=20+0.1 x\) gives the amount of oil in millions of barrels per day that will be supplied to a small country and the function \(D=30-0.5 x\) gives the demand for oil in millions of barrels per day, where \(x\) is the price of oil in dollars per barrel. The president worries if the supply is less than 22 million barrels per day or if demand is less than 15 million barrels per day. For what values of \(x\) does the president worry? Use interval notation.

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Solve each equation. Practice combining some steps. Look for more efficient ways to solve each equation. $$$5 x+1=0$$
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