Solving Linear Inequalities by Factoring

Linear inequalities are mathematical statements that compare two linear expressions using inequality symbols like <, >, ≤, or ≥.

Solving linear inequalities involves isolating the variable on one side of the inequality symbol. One method for solving linear inequalities is factoring.

Factoring allows you to break down an inequality into smaller pieces that are easier to solve.

In this article, we will discuss solving linear inequalities by factoring.

Steps for Solving Linear Inequalities by Factoring

To solve linear inequalities by factoring, we follow these steps:

• Factor the linear expression on one or both sides of the inequality.
• Set each factor equal to zero and solve for the variable.
• The solutions to the inequality are the values of the variable that make one or more of the factors greater than or less than zero, depending on the inequality sign.

Rules for Factoring out Terms

When factoring to solve inequalities, there are some important rules to follow:

• You can only factor out positive terms. Negative terms cannot be factored out.
• If you factor out a negative term, you must flip the inequality symbol.
• Factors that are variables cannot be divided out. Only numerical coefficients can be divided out.

Solved Examples Linear Inequalities by Factoring

Example

Solve the following linear inequality:

x + 2 > 5

Solution

Subtract 2 from both sides:

x > 5 – 2

Simplify,

x > 3

The solution to the inequality is all values of x that are greater than 3.

Example

Solve the following linear inequality:

-2x + 3 < 0

Solution

Subtract 3 from both sides,

-2x < 0 – 3

Simplify,

-2x < -3

Divide both sides by -2. Remember to change the direction of the inequality since we are dividing by a negative number.

x > 1.5

The solution to the inequality is all values of x that are greater than 1.5.

Example

Solve the following linear inequality

x² – 5x + 6 < 0

Solution

Factor the quadratic expression on the left side:

(x – 3)(x – 2) < 0

Set each factor equal to zero and solve for the variable:

x – 3 = 0 , x – 2 = 0

x = 3 , x = 2

Our solutions are x = 3 and x = 2. However, we need to be careful because we are dividing by zero in the quadratic expression. We can use a test point to check which values of x make the expression less than zero.

Let’s try x = 1.

(x – 3)(x – 2) < 0

(1 – 3)(1 – 2) < 0

(-2)(-1) < 0

2 < 0

This is not true, so x = 1 is not a solution.

Let’s try x = 4.

(x – 3)(x – 2) < 0

(4 – 3)(4 – 2) < 0

(1)(2) < 0

2 < 0

This is also not true, so x = 4 is not a solution.

Therefore, the only solution to the inequality is x = 2.

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