Problem 39 - Algebra for College Students

Question

Determine which of the ordered pairs \((1,3),(-2,5)\) \((-6,-4),\) and \((7,-8)\) satisfy each compound or absolute value inequality. $$y>-x+1 \text { or } y>4 x$$

Step-by-Step Solution

Answer

The pairs that satisfy the inequalities are (1, 3), (-2, 5), and (-6, -4).

Step 1 - Understand the inequalities

We need to determine which of the given ordered pairs satisfy either the inequality \(y > -x + 1\) or \(y > 4x\). This means we will check each pair against both of these conditions.

Step 2 - Evaluate inequality \(y > -x + 1\)

Check each ordered pair by substituting the values into \(y > -x + 1\):1. For (1, 3): \(3 > -1 + 1\) which simplifies to \(3 > 0\) (True)2. For (-2, 5): \(5 > -(-2) + 1\) which simplifies to \(5 > 3\) (True)3. For (-6, -4): \(-4 > -(-6) + 1\) which simplifies to \(-4 > 7\) (False)4. For (7, -8): \(-8 > -7 + 1\) which simplifies to \(-8 > -6\) (False)

Step 3 - Evaluate inequality \(y > 4x\)

Check each ordered pair by substituting the values into \(y > 4x\):1. For (1, 3): \(3 > 4 * 1\) which simplifies to \(3 > 4\) (False)2. For (-2, 5): \(5 > 4 * (-2)\) which simplifies to \(5 > -8\) (True)3. For (-6, -4): \(-4 > 4 * (-6)\) which simplifies to \(-4 > -24\) (True)4. For (7, -8): \(-8 > 4 * 7\) which simplifies to \(-8 > 28\) (False)

Step 4 - Determine the pairs that satisfy at least one inequality

We need to check if each pair satisfies at least one of the inequalities. Summarize the results:1. (1, 3) satisfies \(y > -x + 1\)2. (-2, 5) satisfies both \(y > -x + 1\) and \(y > 4x\)3. (-6, -4) satisfies \(y > 4x\)4. (7, -8) satisfies neither inequality

Key Concepts

Absolute Value Inequalities

Absolute value inequalities are expressions that involve the absolute value of a variable or an expression containing a variable. The absolute value of a number is its distance from zero on the number line, regardless of direction. These inequalities come in two major forms:

• \(|A| < B\) means that \(-B < A < B\)
• \(|A| > B\) means that \((A < - B) \text{ or } (A > B)\)

Understanding these basic forms will help in solving more complex problems. When dealing with absolute value inequalities, you can translate them into compound inequalities to find the solutions. This can be particularly useful when you need to solve for ranges of values that a variable can take.

Inequality Solutions

Inequality solutions involve determining the values that satisfy an inequality. In the given exercise, you had to check if the ordered pairs (x, y) satisfy either \(y > -x + 1\) or \(y > 4x\). This means:

• For \(y > -x + 1\), you consider all y-values greater than \(-x + 1\).
• For \(y > 4x\), you look at all y-values greater than \(4x\).

You solve these inequalities by substituting the values of the ordered pairs and checking whether the resulting statements are true or false. For example, substituting (1, 3) into \(y > -x + 1\):

• \((3 > -1 + 1)\) simplifies to \(3 > 0\), which is true.

This process helps you determine which ordered pairs are valid solutions. To find the compound inequality solutions, you combine the results from each individual inequality to see which pairs satisfy at least one condition.

Substituting Values

Substituting values into inequalities is a crucial step in finding solutions. It involves replacing the variables in the inequality with specific numeric values and then solving it. Let’s see this in action with the ordered pairs and inequalities from the exercise.
First, you take each pair \( (x, y) \) and substitute the x and y values into the inequality expressions:

• For \( (1, 3): \) Substitute into \( y > -x + 1 \) and \( y > 4x \)
  
  • \((3 > -1 + 1) \) simplifies to \( 3 > 0 \) (True)
  • \((3 > 4*1) \) simplifies to \(3 > 4 \) (False)
• For \( (-2, 5): \) Substitute into \( y > -x + 1 \) and \( y > 4x \)
  
  • \((5 > -(-2) + 1) \) simplifies to \(5 > 3 \) (True)
  • \((5 > 4*(-2)) \) simplifies to \(5 > -8 \) (True)

This substitution process allows you to see clearly whether each pair meets the inequality conditions. Accurately substituting the values into each inequality and solving helps confirm which ordered pairs are correct solutions.