Envision Math Accelerated Grade 7 Volume 1 Chapter 6 Solving Problems Using Equations And Inequalities
Question. How can you solve real-world and mathematical problems with numerical and algebraic equations and inequalities?
Given
Statement
To find/solve
How can you solve real-world and mathematical problems with numerical and algebraic equations and inequalities?
The equation shows the relationship between variables and other quantities in a situation which involves different operations on numbers.
The left side of an equation must have the same value with the right side to make the equality true.
The variable will also indicate balance in the relationship of the equation.
The left side of an equation must have the same value with the right side to make the equality true.
The left side of an equation must have the same value with the right side to make the equality true.
We need to research the need for safe, clean water in developing countries.
Based on the research, we need to determine the type, size, and cost of a water filtration system needed to provide clean, safe water to a community.
Also, we have to develop a plan to raise money to purchase the needed filtration system.
Reverse osmosis systems are the most effective filters for drinking water.
Many of them feature seven or more filtration stages along with the osmosis process that makes them.
Effective at moving 99 percent of contaminants from water, including chemicals such as chlorine, heavy metals, pesticides, and herbicides.
Filtered water reduces corrosion and improves PH levels also extending the life of household fixtures.
It costs around $30 per square foot.
We can organize community parties to fetch donations for raising money for the purchase of the needed filtration system.
Reverse osmosis systems are the most effective filters for drinking water. It helps us in getting safe, clean water in developing countries.
A property of equality states that performing the same operation on both sides of an equation will keep the equation true.
Properties that state that performing the same operation on both sides of an equation will keep the equation true are called properties of equality.
The inverse relationship is involved in addition and subtraction because they can “undo” each other.
Addition and subtraction have a(n) inverse relationship because they can “undo” each other.
Like terms are terms that have same the variables and powers
Example:
5x+10x
Terms that have the same variable are called like terms.
Question. Find the value of x to solve the given equation x + 9.8 = 1.2
We need to use properties to solve the given equation for x
The given equation is x + 9.8 = 14.2
The given equation is x + 9.8 = 14.2
Solving it, we get
x + 9.8 = 14.2
x + 9.8 − 9.8 = 14.2 − 9.8
x = 4.4
The value of x=4.4
Question. Find the value of x solve the given equation 14x = 91.
We need to use properties to solve the given equation for x
The given equation is 14x = 91
The given equation is 14x = 91
Solving it, we get
14x = 91
\(\frac{14x}{14}\)=\(\frac{91}{14}\)
x = 13
The value of x = 13
Question. Find the value of x solve the given equation \(\frac{1}{3}x\)
We need to use properties to solve the given equation for x
The given equation is \(\frac{1}{3}x\) = 24
The given equation is \(\frac{1}{3}\)x = 24
Solving it, we get
\(\frac{1}{3}\)x = 24
3×\(\frac{1}{3}\)x = 3 × 24
x = 3 × 24
x = 72
The value of x = 72
Question. Solve the expression and combine like terms \(\frac{1}{4}+\frac{1}{4}m-\frac{2}{3}k+\frac{5}{9}m\).
We need to combine like terms in the given expression.
The given expression is \(\frac{1}{4} k+\frac{1}{4} m-\frac{2}{3} k+\frac{5}{9} m\)
The given is, \(\frac{1}{4} k+\frac{1}{4} m-\frac{2}{3} k+\frac{5}{9} m\)
Combining the like terms, we get
\(\frac{1}{4} k+\frac{1}{4} m−\frac{2}{3} k+\frac{5}{9} m\)= \(\frac{1}{4} k−\frac{2}{3} k + \frac{1}{4} m+\frac{5}{9} m\)
= k(\(\frac{1}{4}\)−\(\frac{2}{3}\)) + m( \(\frac{1}{4}\) + \(\frac{5}{9}\))
= k(\(\frac{3−8}{12}\)) + m(\(\frac{9+20}{36}\))
= k (\(\frac{−5}{12}\)) + m (\(\frac{29}{36}\))
=\(\frac{−5}{12}\) k + \(\frac{29}{36}\)m
The expression becomes \(\frac{-5}{12} k+\frac{29}{36} m\)
Question. Solve the expression and combine like terms -4b + 2w + (-4b) + 8w.
We need to combine like terms in the given expression.
The given expression is −4b + 2w + (−4b) + 8w
The given is, −4b + 2w + (−4b) + 8w
Combining the like terms, we get
−4b + 2w + (−4b) + 8w
= −4b + 2w − 4b + 8w
= b(−4−4) + w(2 + 8)
= −8b + 10w
The expression becomes −8b + 10w
Question. Solve the expression and combine like terms 6 – 5z + 8 – 4z + 1.
We need to combine like terms in the given expression.
The given expression is 6−5z + 8 − 4z + 1
The given is 6 − 5z + 8−4z + 1
Combining the like terms, we get
6−5z + 8 − 4z + 1
= 6 + 8 + 1 − 5z − 4z
= 15 − z(5 + 4)
= 15 − 9z
The expression becomes 15 − 9z
Question. A large box of golf balls has more than 12 balls. We need to describe how your inequality represents the situation.
We need to write an inequality that represents the situation
A large box of golf balls has more than 12 balls.
We need to describe how your inequality represents the situation.
Given that a large box of golf balls has more than 12 balls.
Let x be the number of golf balls in a large box.
If it is more than that, it is represented by the symbol <
Thus, the inequality equation will be
12 < x
x > 12
The inequality that represents the situation x > 12
Question. Write the similarities and differences between an equation and an inequality.
We have to write the similarities and differences between an equation and an inequality.
Similarities between an equation and an inequality:
Equation and inequality both use variables when writing and expressing.
Just like the equations, the solution to equality is a value that makes the inequality true.
Both expressions may have different solutions.
Differences:
An equation is a mathematical statement that shows the equal value of two expressions.
While an inequality is a mathematical statement that shows that an expression is lesser than or more than the other.
An equation has only one definite value of a variable while an inequality may have several values because of its range.
Both equations and inequalities show the relation between two expressions. The equation shows that the two expressions are equal while inequalities show that an expression is bigger or smaller than the other.