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Chapter 3: Problem 102

Is the ordered pair a solution to the given inequality? $$ x \leq-5 ;(-6,4) $$

Short Answer

Expert verified
Yes, the ordered pair \((-6, 4)\) is a solution to the inequality \(x \leq -5\).

Step by step solution

01

Identify the inequality

The given inequality is \( x \leq -5 \). Our task is to check whether the ordered pair \((-6, 4)\) satisfies this inequality.
02

Substitute the x-value

From the ordered pair \((-6, 4)\), identify the \(x\) value, which is \(-6\). Substitute \(x = -6\) into the inequality \(x \leq -5\).
03

Evaluate the inequality

After substitution, the inequality becomes \(-6 \leq -5\). Evaluate whether this statement is true or false.
04

Conclusion

Since \(-6 \leq -5\) is a true statement, the ordered pair \((-6, 4)\) is indeed a solution to the inequality \(x \leq -5\).

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Key Concepts

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

Understanding Ordered Pairs
In mathematics, an ordered pair refers to a pair of elements arranged in a specific sequence, typically written as \((x, y)\). This concept is crucial in identifying specific points on a coordinate plane.
In the context of inequalities, the ordered pair helps to determine solutions by using its components.
The first element, \(x\), often represents the horizontal position, while the second element, \(y\), represents the vertical position.
When working with inequalities, our main focus is usually on the \(x\) value of the ordered pair.
This value is substituted into the inequality to verify if the given condition holds true.
Finding the Solution of an Inequality
An inequality involves a mathematical expression using symbols like \(\leq, \geq, <, >\). Unlike equations, inequalities identify a range of possible solutions.
For example, consider the inequality \(x \leq -5\).
This means any value of \(x\) less than or equal to \(-5\) satisfies the inequality.
Ordered pairs play a role in determining solutions by offering specific \(x\) values to test.

  • If the \(x\) value substituted into an inequality upholds the relationship, the ordered pair is a solution.
  • If it doesn鈥檛, then it's not a solution.
Understanding this process provides clarity when working with problems involving inequalities.
Using the Substitution Method
The substitution method is a simple yet powerful strategy for finding whether an ordered pair is a solution to an inequality.
Here's how it works:
  • First, identify the \(x\) value from the given ordered pair, such as \((-6, 4)\).
  • Substitute this \(x\) value into the inequality.
  • Evaluate the statement to see if it's true or false.
For instance, substituting \(x = -6\) into \(x \leq -5\) results in \(-6 \leq -5\).
This evaluation is true, confirming that the ordered pair \((-6, 4)\) satisfies the inequality.

The substitution method offers a straightforward approach to tackle problem-solving with inequalities.

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Most popular questions from this chapter

Find the equation of the line in slope-intercept form. Parallel to \(y=1\) and passing through (4,-1) .

Find the intercepts and graph them. $$ -12 x+23 y=2 $$

Find the equation of a line, given the slope and a point on the line. $$ m=-3 / 4 ;(2,-3) $$

Find the equation of the line given two points on the line. (-6,12) and (3,-3)

Determine whether the given point is a solution. $$ y=-35 x-2 ;(10,-8) $$
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