6.1 Linear Equations & Inequalities

6.1 Linear Equations & Inequalities

Algebra forms the backbone of the Quantitative section of the Saudi Arabia General Aptitude Test (GAT), also known as Qudurat. Within algebra, linear equations and inequalities represent some of the most frequently tested concepts. These problems evaluate your ability to manipulate variables, solve single-variable equations, work with systems of equations, translate word problems into algebraic expressions, and handle linear inequalities.

Solving Single-Variable Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The standard form is:

ax + b = 0

To solve for x, you must isolate the variable on one side of the equation. This is achieved by performing inverse operations in a balanced manner:

1. Simplify: Expand any parentheses and combine like terms on both sides of the equation.
2. Group: Move all variable terms to one side (usually the left) and all constant terms to the other side (usually the right) by adding or subtracting.
3. Isolate: Multiply or divide to solve for the variable, ensuring its coefficient is positive 1.

Worked Example: Basic Linear Equation

Solve for x in the equation: 4(x - 3) = 2x + 10

• Step 1: Distribute the 4 on the left side: 4x - 12 = 2x + 10
• Step 2: Subtract 2x from both sides to group the x-terms: 2x - 12 = 10
• Step 3: Add 12 to both sides to isolate the constant terms: 2x = 22
• Step 4: Divide both sides by 2 to find x: x = 11

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Systems of Linear Equations & Qiyas Shortcuts

A system of linear equations consists of two or more equations with the same set of variables. On the GAT, you will typically deal with two equations and two variables (usually x and y). While standard algebraic methods—substitution and elimination—are useful, the GAT is a highly timed exam (averaging less than 60 seconds per question). Therefore, Qiyas frequently designs systems that can be solved using rapid shortcuts.

Standard Method 1: Substitution

Solve one equation for a single variable, then substitute that expression into the other equation. If: y = 2x - 1 3x + 2y = 12

Substitute y in the second equation: 3x + 2(2x - 1) = 12 3x + 4x - 2 = 12 7x = 14 => x = 2

Then substitute back to find y: y = 2(2) - 1 = 3

Standard Method 2: Elimination

Align the equations and add or subtract them to eliminate one variable. If: 3x + y = 14 x - y = 2

Add the equations directly: (3x + x) + (y - y) = 14 + 2 4x = 16 => x = 4

Substitute x to find y: 4 - y = 2 => y = 2

The Qiyas "Combination" Shortcut

Qiyas questions often ask for the value of a combined expression, such as x + y, x - y, or 3x + 3y, rather than the individual values of x and y. Solving for x and y separately is a waste of time. Look for ways to add or subtract the equations directly to get the desired expression in one step.

Worked Example: Qiyas Shortcut

If 4x + 3y = 18 and 3x + 4y = 17, what is the value of x + y?

If you try to solve this system using elimination or substitution, you will deal with tedious fractions. Instead, notice the coefficients. Simply add the two equations together: (4x + 3y) + (3x + 4y) = 18 + 17 7x + 7y = 35

Factor out the common term (7): 7(x + y) = 35

Divide both sides by 7: x + y = 5

In just three simple steps, without finding x or y individually, you have the exact answer.

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Linear Inequalities and the Negative Sign Rule

Linear inequalities use symbols like < (less than), > (greater than), <= (less than or equal to), and >= (greater than or equal to). Solving inequalities follows the same rules as solving linear equations, with one critical exception: When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

For example: -2x < 8

Dividing both sides by -2 requires flipping the sign from < to >: x > -4

Inequality Concepts Table

Worked Example: Solving an Inequality

Solve the inequality for x: -3(x - 2) >= 12

• Step 1: Distribute the -3: -3x + 6 >= 12
• Step 2: Subtract 6 from both sides: -3x >= 6
• Step 3: Divide both sides by -3 and flip the inequality sign: x <= -2

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Absolute Value Equations & Inequalities

The absolute value of a number, written as |x|, represents its distance from zero on the number line. Because distance is always non-negative, the absolute value is always positive or zero.

• Equations: If |x| = a (where a >= 0), then x = a or x = -a.
• Inequalities (Less Than): If |x| < a, then -a < x < a.
• Inequalities (Greater Than): If |x| > a, then x > a or x < -a.

Worked Example: Absolute Value Inequality

Solve for x: |2x - 5| < 7

• Step 1: Set up the compound inequality: -7 < 2x - 5 < 7
• Step 2: Add 5 to all three parts: -2 < 2x < 12
• Step 3: Divide all parts by 2: -1 < x < 6

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Translating Word Problems into Linear Equations

The GAT contains many word problems that must be translated into algebraic form to solve. Success depends on recognizing key verbal cues.

Algebraic Translation Key

Worked Example: Age Problem

A mother is currently 4 times older than her daughter. In 6 years, the mother's age will be 3 times her daughter's age at that time. How old is the daughter now?

• Step 1: Define variables. Let the daughter's current age be d. The mother's current age is 4d.
• Step 2: Express their ages in 6 years: Daughter: d + 6 Mother: 4d + 6
• Step 3: Set up the equation based on the future relationship: 4d + 6 = 3(d + 6)
• Step 4: Solve the equation: 4d + 6 = 3d + 18 Subtract 3d from both sides: d + 6 = 18 Subtract 6 from both sides: d = 12 The daughter is currently 12 years old (and the mother is 48 years old).

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Qiyas Traps and Pitfalls to Avoid

1. The Inequality Sign Flip: The absolute most common error is forgetting to reverse the inequality sign when dividing or multiplying by a negative number. Always check if you divided by a negative coefficient.
2. Solving for the Wrong Variable/Expression: Read the final line of the question carefully. If the question asks for "2x + 5" and you solve x = 4, do not select 4. Calculate 2(4) + 5 = 13.
3. Double Age Shifts: In age word problems, when moving forward or backward in time, you must add or subtract years from both individuals. Many candidates write 4d + 6 = 3d instead of 4d + 6 = 3(d + 6).