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Understanding inequalities and equations
By MathPlanetVideos
Margaret is paying $1,500 in rent for her apartment every month. Margaret won $12,000 in Vegas and wondering how many rent she can afford to pay with this winning?
Difference of Two Cubes | MathHelp.com
By MathHelp.com
To solve the given system of inequalities, we start by graphing the associated equation for each inequality. In other words, we graph y equals -1/5 x +1 and y equals 3x + 2. So, for the first inequality, we start with our y-intercept of positive 1, up 1 unit on the y-axis. From there, we take our slope of -1/5, so we go down 1 and to the right 5, and plot a second point. Now, notice that our inequality uses a “less than” sign. This means that we draw a dotted line connecting the points, rather than a solid line. It’s important to understand that if we have a greater than sign or a less than sign, we use a dotted line, and if we have a greater than or equal to sign or a less than or equal to sign, we use a solid line. Pay close attention to this idea when drawing your lines. Students often carelessly use a solid line when they should use a dotted one, and vice-versa. Next, let’s take a look at our second inequality, which has a y-intercept of positive 2, up 2 units on the y-axis. From there, we take our slope of 3, or 3 over 1, so we go up 3 and to the right 1, and plot a second point. And notice that this inequality uses a “greater than or equal to” sign, so we connect the points with a solid line, rather than a dotted line. Next, we need to determine which side of each of these lines to shade on the graph. To determine which side of our first line to shade, we use a test point on either side of the first line. The easiest test point to use is (0, 0), so we plug a zero into our first inequality for both x and y, and we have 0 is less than -1/5 times 0 + 1, which simplifies to 0 is less than 0 + 1, or 0 is less than 1. Notice that 0 is less than 1 is a true statement. This means that our test point, (0, 0), is a solution to the first inequality, so we shade in the direction of (0, 0) along our first boundary line. Next, we determine which side of our second line to shade by using a test point on either side of the second line, such as (0, 0). Plugging a zero into our second inequality for both x and y, we have 0 equal to 3 times 0 + 2, which simplifies to 0 equal to 0 + 2, or 0 equal to 2. Notice that 0 equal to 2 is a false statement. This means that our test point, (0, 0), is a not solution to the inequality, so we shade away from (0, 0) along our second boundary line. Finally, it’s important to understand that the solution to this system of inequalities is represented by the part of the graph where the two shaded regions overlap, which in this case is in the lower left. In other words, any point that lies in this part of the graph is a solution to the given system of inequalities. Note that the points along the dotted boundary line of this region are not solutions to the system, but the points along the solid boundary line of this region are solutions to the system.
Work Word Problems | MathHelp.com
By MathHelp.com
To solve a polynomial inequality, like the one shown here, our first step is to write the corresponding equation. In other words, we simply change the inequality sign to an equals sign, and we have x^2 – 3 = 9 – x. Next, we solve the equation. Since we have a squared term, we first set the equation equal to 0. So we move the 9 – x to the left side by subtracting 9 and adding x to both sides of the equation. This gives us x^2 + x – 12 = 0. Next, we factor the left side as the product of two binomials. Since the factors of negative 12 that add to positive 1 are positive 4 and negative 3, we have x + 4 times x – 3 = 0. So either x + 4 = 0 or x – 3 = 0, and solving each equation from here, we have x = -4, and x = 3. Now, it’s important to understand that the solutions to the equation, -4 and 3, represent what are called the “critical values” of the inequality, and we plot these critical values on a number line. However, notice that our original inequality uses a greater than sign, rather than greater than or equal to sign, so we use open dots on our critical values of -4 and positive 3. Remember that ‘greater than’ or ‘less than’ means open dot, and ‘greater than or equal to’ or ‘less than or equal to’ means closed dot. Now, we can see that our critical values have divided the number line into three separate intervals: less than -4, between -4 and 3, and greater than 3. And here’s the important part. Our next step is to test a value from each of the intervals by plugging the value back into the original inequality to see if it gives us a true statement. So let’s first test a value from the “less than -4” interval, such as -5. If we plug a -5 back in for both x’s in the original inequality, we have -5 squared – 3 greater than 9 minus a -5, which simplifies to 25 – 3 greater than 9 + 5, or 22 greater than 14. Since 22 greater than 14 is a true statement, this means that all values in the interval we’re testing are solutions to inequality, so we shade the interval. Next, we test a value from the “between -4 and 3” interval, such as 0. If we plug a 0 back in for both x’s in the original inequality, we have 0 squared – 3 greater than 9 – 0, which simplifies to 0 – 3 greater than 9, or -3 greater than 9. Since -3 greater than 9 is a false statement, this means that all values in the interval we’re testing are not solutions to inequality, so we don’t shade the interval. Next, we test a value from the “greater than 3” interval, such as 4. If we plug a 4 back in for both x’s in the original inequality, we have 4 squared – 3 greater than 9 – 4, which simplifies to 16 – 3 greater than 5, or 13 greater than 5. Since 13 greater than 5 is a true statement, this means that all values in the interval we’re testing are solutions to inequality, so we shade the interval. Finally, we write the answer that’s shown on our graph in set notation. The set of all x’s such that x is less than -4 or x is greater than 3.
Modeling with one-variable equations and inequalities
By Khan Academy
Sal models a context that concerns the number and price of pizza slices. The model turns out to be a rational equation.
Modeling with one-variable equations and inequalities
By Khan Academy
Sal models a context that concerns a candy vending machine. The model turns out to be a quadratic inequality.
Modeling with one-variable equations and inequalities
By Khan Academy
Sal models a context that concerns a bank savings account. The model turns out to be an exponential equation.
Absolute value inequalities | Linear equations | Algebra I | Khan Academy
By Khan Academy
This video lecture series on Worked Examples in Algebra from Khan Academy includes Solving Equations, Solving Word Problems, Solving for a variable, Absolute Value and Number Lines, Patterns in Sequences, Functional Relationships, Domain and Range, Rate Problems, Linear Functions, Slope of a Line, X and Y intercepts, Equation of a Line, Parallel Lines, Perpendicular Lines, Solving Inequalities and more...
Reasoning with Equations and Inequalities
By Khan Academy
Sal solves the equation 4x^2+40x-300=0 by completing the square.265
Proof that rational times irrational is irrational | Algebra I | Khan Academy
By Khan Academy
Algebra I on Khan Academy: Algebra is the language through which we describe patterns. Think of it as a shorthand, of sorts. As opposed to having to do something over and over again, algebra gives you a simple way to express that repetitive process. It's also seen as a "gatekeeper" subject. Once you achieve an understanding of algebra, the higher-level math subjects become accessible to you. Without it, it's impossible to move forward. It's used by people with lots of different jobs, like carpentry, engineering, and fashion design. In these tutorials, we'll cover a lot of ground. Some of the topics include linear equations, linear inequalities, linear functions, systems of equations, factoring expressions, quadratic expressions, exponents, functions, and ratios. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
Grade 7 Math - Modeling Using Equations or Inequalities
By Lumos Learning
Using the Lumos Study Programs, parents and educators can reinforce the classroom learning experience for children and help them succeed at school and on the standardized tests. Lumos books, dvd, eLearning and tutoring are used by leading schools, libraries and thousands of parents to supplement classroom learning and improve student achievement in the standardized tests.
Functions and Graphs
By The Organic Chemistry Tutor
This precalculus provides a basic introduction into functions and graphs. It contains plenty of examples and multiple choice practice problems.
01 - The Distance Formula, Pythagorean Theorem & Midpoint Formula - Part 1 (Calculate Distance)
By Math and Science
Quality Math And Science Videos that feature step-by-step example problems!
08 - Solving Exponential Equations - Part 1 - Solve for the Exponent
By Math and Science
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Static Friction and Kinetic Friction Physics Problems With Free Body Diagrams
By The Organic Chemistry Tutor
This physics video tutorial provides a basic introduction into kinetic friction and static friction. It contains plenty of examples and physics problems that asks you to calculate the acceleration using newton's laws of motion. The static frictional force is equal to the applied force up to a maximum value. The kinetic friction force is a constant value that depends on the interaction between the horizontal surface and the object. All frictional forces is dependent on the normal of the object. You should draw a free body diagram for each practice problem if you wish to make it a lot easier.
15 - What is a Logarithm (Log x) Function? (Calculate Logs, Applications, Log Bases)
By Math and Science
Quality Math And Science Videos that feature step-by-step example problems!