EQUATIONS AND INEQUALITIES

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Equations and inequalities are widely used in the industry to solve a variety of problems and make important decisions. Here are some examples of how equations and inequalities are used in different industrial applications:

Manufacturing: In manufacturing, equations are used to calculate the optimal production rates and minimize the cost of production. Inequalities are used to ensure that the production process stays within safety limits, such as maximum temperature or pressure.
Engineering: Equations are used extensively in engineering to design and optimize systems, such as determining the forces acting on a structure or the optimal design of a machine. Inequalities are used to ensure that the system operates safely, such as limiting the stress or strain on materials.
Logistics and supply chain management: Equations and inequalities are used to optimize the flow of goods and services in logistics and supply chain management. For example, they can be used to minimize transportation costs while meeting delivery deadlines or to allocate resources optimally among different production centers.
Finance and accounting: Equations and inequalities are used in finance and accounting to calculate interest rates, investment returns, and to optimize financial decisions. Inequalities are used to set budget constraints and ensure that financial decisions comply with legal and regulatory requirements.
Quality control: Equations and inequalities are used in quality control to determine whether a product meets certain standards. For example, they can be used to test whether a component is within a specified tolerance limit or whether a product is free of defects.

How do you know if the given is an equations or inequalities?

An equation and an inequality are both mathematical statements involving one or more variables. The main difference between the two is that an equation asserts that two expressions are equal, while an inequality asserts that two expressions are not equal and specifies the relationship between them.

To determine whether a given statement is an equation or an inequality, look for the presence of an equal sign (=) or any of the inequality signs (<, >, ≤, ≥).

If there is an equal sign present, then the statement is an equation. For example, "2x + 3 = 7" is an equation.

If there is an inequality sign present, then the statement is an inequality. For example, "2x + 3 < 7" is an inequality.

It's important to note that some mathematical statements can be both an equation and an inequality, depending on the values of the variables involved. For example, "x + 3 = 7" is an equation, but it can also be written as "x < 4" or "x ≤ 4", which are both inequalities.

Solved problems with Solutions

Equations

1. Solve for x: 3x + 7 = 22

Solution:

$3x + 7 = 22$

$3x + 7 {\color{Red} -7}= 22{\color{Red} -7}$

$3x = 15$

$(3x = 15)*{\color{Red} \frac{1}{3}}$

$x=5$ , Answer

2, Solve for x: 2x + 5 = 13.

Solution:

$2x + 5 = 13$

$2x + 5{\color{Red} =5} = 13{\color{Red} =5}$

$2x = 8$

$(2x = 8)*{\color{Red} \frac{1}{2}}$

$x = 4$ , Answer

3. Solve for y in the equation 3(y + 2) = 21.

Solution:

$3(y + 2) = 21$

$3y + 6 = 21$

$3y + 6{\color{Red} -6} = 21{\color{Red} -6}$

$3y = 15$

$(3y = 15)*{\color{Red} \frac{1}{3}}$

$y = 5$ ,Answer

4. Solve for x in the equation 4x - 7 = 9.

Solution:

$4x-7=9$

$4x-7{\color{Red} +7}=9{\color{Red} +7}$

$4x=16$

$(4x=16)*{\color{Red} \frac{1}{4}}$

$\frac{4x}{{\color{Red} 4}}=\frac{16}{{\color{Red} 4}}$

$x=4$ ,Answer

4. Solve for y in the equation below.

$\frac{2y}{3} + 4 = 10$

Solution:

$\frac{2y}{3} + 4 = 10$

$\frac{2y}{3} + 4 {\color{Red} - 4} = 10 {\color{Red} - 4}$

$\frac{2y}{3} = 6$

$(\frac{2y}{{\color{Red} 3}} = 6)*{\color{Red} 3}$

$2y=18$

$({\color{Red} 2}y=18)*\frac{1}{{\color{Red} 2}}$

$y=9$ ,Answer

5. Solve for x in the equation 2x - 3 = 7x + 4.

Solution:

$2x - 3 = 7x + 4$

$2x - 3{\color{Red} +3} = 7x + 4{\color{Red} +3}$

$2x = 7x + 7$

or flip it into

$7x + 7=2x$

$7x + 7{\color{Red} -7}=2x{\color{Red} -7}$

$7x =2x{\color{Red} -7}$

$7x{\color{Blue} -2x} =2x{\color{Blue} -2x}{\color{Red} -7}$

$5x=-7$

$({\color{Red} 5}x=-7)*\frac{1}{{\color{Red} 5}}$

$x=-\frac{7}{5}=-1\frac{2}{5}$ , Answer

Solved problems with Solutions

Inequalities

1. Solve for x in the inequality 2x + 5 < 11.

Solution:

$2x + 5 < 11$

$2x + 5 {\color{Red} - 5} < 11 {\color{Red} - 5}$

$2x < 6$

$({\color{Red} 2}x < 6)*\frac{1}{{\color{Red} 2}}$

$x < 3$ , Answer

2. Solve for x in the inequality 3x - 7 > 8.

Solution:

$3x - 7 > 8$

$3x - 7{\color{Red} +7} > 8{\color{Red} +7}$

$3x > 15$

$({\color{Red} 3}x > 15)*\frac{1}{{\color{Red} 3}}$

$x > 5$ , Answer

3. Solve for x in the inequality -2x + 6 ≤ 10.

Solution:

$-2x+6\leq 10$

$-2x+6{\color{Red} -6}\leq 10{\color{Red} -6}$

$2x\leq 4$

$({\color{Red} 2}x\leq 4)*\frac{1}{{\color{Red} 2}}$

$x\leq 2$ , Answer

4. Solve for x in the inequality 5x + 3 > 4x + 8.

Solution:

$5x + 3 > 4x + 8$

$5x{\color{Blue} -4x} + 3{\color{Red} -3} > 4x{\color{Blue} -4x} + 8{\color{Red} -3}$

$x > 5$ ,Answer

5. Solve for x in the inequality 3x - 7 ≥ 8.

Solution:

$3x - 7 \geq 8$

$3x - 7{\color{Red} +7} \geq 8{\color{Red} +7}{\color{Red} }$

$3x =15$

$({\color{Red} 3}x =15)*\frac{1}{{\color{Red} 3}}$

$x \geq 5$

Exercises:

Part 1:

Solve for the value of x.

Equation: 4x + 5 = 21
Inequality: 2y - 3 > 7
Equation: 6a - 8 = 22
Inequality: 3x + 2 ≤ 10
Equation: 2b + 4 = 16
Inequality: 5y - 7 < 13
Equation: 3x + 7 = 16
Inequality: 4a - 2 ≥ 10
Equation: 5y - 3 = 22
Inequality: 2x + 1 > 5

Part 2:

Solve for the value of x and identify if the given is an equation or inequality.

3x + 7 = 16
2x - 5 < 8
2y - 4 = 10
4x + 3 ≥ 15
5a + 2 = 17
3x - 4 > 7
4b - 8 = 12
6y + 2 ≤ 20
2x + 5 = 3x - 1
5a - 7 > 3a + 5

Things to remember about equations and inequality.

1. Equations and inequalities are mathematical expressions that describe the relationship between two quantities. Equations contain an equal sign (=) and express that the two sides are equivalent, while inequalities contain symbols such as <, >, ≤, or ≥ and express that the two sides are not equal.

2. When solving an equation or inequality, it is important to perform the same operation on both sides of the equation or inequality in order to maintain balance. This means that if you add, subtract, multiply, or divide one side of the equation or inequality by a certain number or variable, you must also perform the same operation on the other side. Additionally, when multiplying or dividing by a negative number, you must flip the inequality symbol to maintain the same relationship between the two sides.

DMG