How to Solve Linear Equation and Inequality Word Problems?

In this article, you will learn how to solve Linear equations and Inequality Word Problems in a few simple steps.

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Step-by-step guide to solving linear equations and Inequality Word Problems

Linear equations are relations that describe a straight line. In linear equations, the quantities \(x, y\) don’t have any power or radical. The general form of this type of equation is as follows: \(ax+b=0\) where \(a\) and \(b\) are constant numbers \(a≠0\) and \(x\) is also unknown.

Linear inequalities are very similar to linear equations, and in them, the power of the variables is not more than \(1\). Solving linear inequalities is very similar to solving linear equations. The most important thing in solving linear inequalities is to remember that when you multiply or divide the sides of an equation by a negative number, you invert the inequality symbol. You also need to remember that in a linear inequality, you do not have just one answer, but a whole set of answers, and the number of answers in a linear inequality can be infinite.

Word problems refer to problems that have a narrative mode and, instead of giving you a linear equation or linear inequality directly to solve, they bring you a narrative so that you can discover the equation or inequality yourself.

To solve a linear equation or linear inequality word problem, you can use these strategies:

• Read through the entire problem.
• Highlight the important information and keywords that you need to solve the problem:

1. Replace and, more than, exceeded by, with a \((+)\) sign.
2. Replace less than, less, subtract with a \((-)\) sign.
3. Replace of times as much with a \((\times)\) sign.
4. Replace twice with \((2\times)\).
5. Replace divided by with \((÷)\).
6. Replace half as much with \((\frac{1}{2}\times)\).
7. Replace the verbs is, are with a \((=)\) sign.

• Identify your variables.
• Write the inequality or equation.
• Solve.
• Write your answer in a complete sentence.

Solving Linear Equations and Inequality Word Problems Example 1:

When \(233\) is added to a number, the result is \(65\) more than \(5\) times the number. Find the number.

Solution: Let \(x\) be the number. Then, write the equation: \(233+x=5x+65→233-65=5x-x→168=4x→x=42\)

Solving Linear Equations and Inequality Word Problems Example 2:

Alice has scored \(95\) points in the first level of a game. To play the third level, she needs more than \(250\) points. To play the third level, how many points should she score in the second level?

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Solution: let \(x\) be the point scored in the second level. She has already had \(95\) points in the first level. Points scored in the second level\(=x\)

Total points in the first two levels\(=x+95\)

Write the inequality: to play the third level, the total points in the first two levels should be more than \(250\). So, we have\(x+95>250\)

Subtract \(95\) from both sides.\((x+95)-95>250-95→x>155\)

So, she has to score more than \(155\) points in the second level.

Solving Linear Equations and Inequality Word Problems Example 3:

An employer recruits experienced \((x)\) and fresh workmen \((y)\) for his firm under the condition that he cannot employ more than \(12\) people. If \(8\) freshmen are recruited, how many experienced men have to be recruited?

Solution: Write the inequality. \(x+y<12\)

Substitute \(8\) for \(y\). \(x+8<12\)

Subtract \(8\) from both sides. \((x+8)-8<12-8→x<4\)

To meet the given condition, no. of freshmen to be recruited can be less than \(4\).

Solving Linear Equations and Inequality Word Problems Example 4:

The sum of a number and \(88\) is greater than the product \(5\) and that number. What are the possible values for the number?

Solution: Let \(x\) be the number. Then, write the inequality. \(x+88>5x→88>4x→22>x\)

Exercises for Solving Linear Equations and Inequality Word Problems

Establish a variable, write an inequality to represent the scenario, and solve.

1. Half of a number, decreased by twenty-two, is at most forty-six. Find the number.
2. The length of a rectangle is treble that of the width. The perimeter of the rectangle is \(48cm\). What is the width of the rectangle?
3. A taxi charges a flat rate of \($4\), plus an additional \($0.55\) per mile. If Sandra has at most \($15\) to spend on the cab ride, how far could she travel?
4. Tickets to a cinema cost \($15\)each, including tax. Mat has \($210\) and wants to get tickets for himself and some friends. How many friends can he invite?

1. \(x=135\)
2. \(x=6\)
3. \(x≤20mile\)
4. \(x=14\)