Problem 35 - 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. \(y>4\) and \(x<1\)

Step-by-Step Solution

Answer

Only the ordered pair (-2, 5) satisfies both conditions.

Step 1: Analyze the inequality conditions

The exercise involves two conditions: the first one is an inequality in terms of y ( y > 4 ) and the second one is an inequality in terms of x ( x < 1 ). We'll check each ordered pair against these conditions.

Step 2: Verify the first ordered pair (1,3)

For the pair (1, 3): 1) Check if y > 4: Here, y = 3. Since 3 is not greater than 4, this pair does not satisfy y > 4 . 2) Check if x < 1: Here, x = 1. Since 1 is not less than 1, this pair does not satisfy x < 1 . Therefore, (1, 3) does not satisfy the compound inequality.

Step 3: Verify the second ordered pair (-2,5)

For the pair (-2, 5): 1) Check if y > 4: Here, y = 5. Since 5 is greater than 4, this pair satisfies y > 4 . 2) Check if x < 1: Here, x = -2. Since -2 is less than 1, this pair satisfies x < 1 . Therefore, (-2, 5) satisfies the compound inequality.

Step 4: Verify the third ordered pair (-6,-4)

For the pair (-6, -4): 1) Check if y > 4: Here, y = -4. Since -4 is not greater than 4, this pair does not satisfy y > 4 . 2) Check if x < 1: Here, x = -6. Since -6 is less than 1, this pair satisfies x < 1 . However, since it doesn't satisfy the first condition, (-6, -4) does not satisfy the compound inequality.

Step 5: Verify the fourth ordered pair (7,-8)

For the pair (7, -8): 1) Check if y > 4: Here, y = -8. Since -8 is not greater than 4, this pair does not satisfy y > 4 . 2) Check if x < 1: Here, x = 7. Since 7 is not less than 1, this pair does not satisfy x < 1 . Therefore, (7, -8) does not satisfy the compound inequality.

Step 6: Conclusion

Reviewing all the checks, only the pair (-2, 5) satisfies both conditions y > 4 and x < 1 .

Key Concepts

absolute value inequalities

Absolute value inequalities involve expressions where the variable is within absolute value symbols, such as \(|x| < 5\). The absolute value represents the distance of a number from zero on a number line, without considering direction. \( |x| < 5 \) translates to all numbers less than 5 units away from zero, including both negative and positive values.

ordered pairs

An ordered pair is a set of two numbers written in a specific order, usually as \((x, y)\). The first number signifies the value on the x-axis, and the second number indicates the value on the y-axis. For instance, in \((1, 3)\), 1 is the x-coordinate and 3 is the y-coordinate. Ordered pairs are crucial in graphing equations and inequalities, as they pinpoint exact locations on a coordinate plane.

inequality conditions

Inequality conditions are statements that describe the relative size or order of two values. They use symbols like \(>\), \(<\), \(eq\), \(\text{≥}\), and \(\text{≤}\). When solving compound inequalities, you must check all conditions. For example, consider the conditions \(y > 4\) and \(< 1\). Both conditions must be true for the ordered pair to satisfy the compound inequality.

graphing inequalities

Graphing inequalities involves shading a region of the coordinate plane that represents all solutions. To graph \(y > 4\), shade above the line \(y = 4\) because it includes all points where y is greater than 4. Similarly, for \(< 1\), shade left of the line \(x = 1\). The intersection of these shaded areas represents the solutions to the compound inequality.