Factor the quadratic expression

Oct 13, 2025 by pynkocean.com 130 views

Solving Inequalities: Finding the Ordered Pair that Satisfies Both Conditions

In mathematics, inequalities are used to describe relationships between variables. When we have two inequalities, we need to find the values of the variables that satisfy both conditions. In this article, we will explore how to solve inequalities and find the ordered pair that makes both inequalities true.

An inequality is a statement that compares two expressions, indicating whether one is greater than, less than, or equal to the other. Inequalities can be written in various forms, including:

Greater than (>): $a > b$
Less than (<): $a < b$
Greater than or equal to (≥): $a ≥ b$
Less than or equal to (≤): $a ≤ b$

Linear inequalities are inequalities that can be written in the form $ax + by < c$ , where $a$ , $b$ , and $c$ are constants. To solve a linear inequality, we can use the following steps:

Isolate the variable: Move all terms containing the variable to one side of the inequality.
Simplify the inequality: Combine like terms and simplify the inequality.
Graph the solution: Plot the solution on a number line or graph.

The first inequality is $y < 3x - 1$ . To solve this inequality, we can isolate the variable $y$ by subtracting $3x$ from both sides:

y - 3x < -1

Next, we can add $3x$ to both sides to simplify the inequality:

y < 3x - 1

This inequality indicates that $y$ is less than $3x - 1$ .

The second inequality is $y ≥ -x + 4$ . To solve this inequality, we can isolate the variable $y$ by adding $x$ to both sides:

y + x ≥ 4

Next, we can subtract $x$ from both sides to simplify the inequality:

y ≥ -x + 4

This inequality indicates that $y$ is greater than or equal to $-x + 4$ .

To find the ordered pair that satisfies both inequalities, we need to find the values of $x$ and $y$ that make both inequalities true. We can do this by graphing the two inequalities on a coordinate plane and finding the intersection point.

To graph the first inequality, $y < 3x - 1$ , we can plot the line $y = 3x - 1$ and shade the region below the line. This represents the values of $y$ that are less than $3x - 1$ .

To graph the second inequality, $y ≥ -x + 4$ , we can plot the line $y = -x + 4$ and shade the region above the line. This represents the values of $y$ that are greater than or equal to $-x + 4$ .

The intersection point of the two inequalities is the point where the two lines intersect. To find the intersection point, we can set the two equations equal to each other:

3x - 1 = -x + 4

Solving for $x$ , we get:

4x = 5

x = \frac{5}{4}

Substituting this value of $x$ into one of the original equations, we get:

y = 3\left(\frac{5}{4}\right) - 1

y = \frac{15}{4} - 1

y = \frac{11}{4}

Therefore, the ordered pair that satisfies both inequalities is $\left(\frac{5}{4}, \frac{11}{4}\right)$ .

In this article, we explored how to solve inequalities and find the ordered pair that makes both inequalities true. We used the steps of isolating the variable, simplifying the inequality, and graphing the solution to solve the two inequalities. We then found the intersection point of the two inequalities and determined the ordered pair that satisfies both conditions.

$The correct answer is $\boxed{\left(\frac{5}{4}, \frac{11}{4}\right)}$.<br/> **Solving Inequalities: A Q&A Guide**$

In our previous article, we explored how to solve inequalities and find the ordered pair that makes both inequalities true. In this article, we will answer some common questions related to solving inequalities.

A: A linear inequality is an inequality that can be written in the form $ax + by < c$ , where $a$ , $b$ , and $c$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c < 0$ , where $a$ , $b$ , and $c$ are constants.

A: To solve a quadratic inequality, you can use the following steps:

Factor the quadratic expression: If possible, factor the quadratic expression into the product of two binomials.
Set each factor equal to zero: Set each factor equal to zero and solve for the variable.
Graph the solution: Plot the solution on a number line or graph.

A: A strict inequality is an inequality that is written with a strict symbol, such as $< $ or $> $. A non-strict inequality, on the other hand, is an inequality that is written with a non-strict symbol, such as $≤ $ or $≥ $.

A: To graph a linear inequality, you can use the following steps:

Plot the line: Plot the line that represents the inequality.
Shade the region: Shade the region that represents the solution to the inequality.

A: The intersection point of two inequalities is the point where the two lines intersect. To find the intersection point, you can set the two equations equal to each other and solve for the variable.

A: To find the ordered pair that satisfies both inequalities, you can use the following steps:

Graph the inequalities: Graph the two inequalities on a coordinate plane.
Find the intersection point: Find the intersection point of the two inequalities.
Determine the ordered pair: Determine the ordered pair that satisfies both inequalities.

A: Some common mistakes to avoid when solving inequalities include:

Not isolating the variable: Failing to isolate the variable can make it difficult to solve the inequality.
Not simplifying the inequality: Failing to simplify the inequality can make it difficult to graph the solution.
Not graphing the solution: Failing to graph the solution can make it difficult to determine the ordered pair that satisfies both inequalities.

In this article, we answered some common questions related to solving inequalities. We covered topics such as linear and quadratic inequalities, strict and non-strict inequalities, graphing linear inequalities, and finding the intersection point of two inequalities. We also discussed some common mistakes to avoid when solving inequalities.

For more information on solving inequalities, you can check out the following resources:

Mathway: A online math problem solver that can help you solve inequalities.
Khan Academy: A online learning platform that offers video lessons on solving inequalities.
Math Open Reference: A online math reference book that offers information on solving inequalities.

The correct answers to the questions are:

Q1: A linear inequality is an inequality that can be written in the form $ax + by < c$ , where $a$ , $b$ , and $c$ are constants.
Q2: To solve a quadratic inequality, you can use the following steps: factor the quadratic expression, set each factor equal to zero, and graph the solution.
Q3: A strict inequality is an inequality that is written with a strict symbol, such as $< $ or $> $.
Q4: To graph a linear inequality, you can plot the line and shade the region that represents the solution.
Q5: The intersection point of two inequalities is the point where the two lines intersect.
Q6: To find the ordered pair that satisfies both inequalities, you can graph the inequalities, find the intersection point, and determine the ordered pair.
Q7: Some common mistakes to avoid when solving inequalities include not isolating the variable, not simplifying the inequality, and not graphing the solution.