Problem 34 - Algebra 1: Concepts and Skills

Question

Solve the inequality. $$ -(2 x+4)>6 $$

Step-by-Step Solution

Answer

The solution to the inequality is x < -5.

Step 1: Negative Sign Removal

To remove the negative sign, multiply the whole inequality by -1. Remember that when we multiply or divide by a negative number, the direction of the inequality changes. Therefore, -(2x + 4) > 6 becomes 2x + 4 < -6.

Step 2: Rearrange the Inequality for constants

The objective is to isolate 'x'. Thus, subtract 4 from both sides of the inequality. This gives: 2x < -6 - 4 which simplifies to 2x < -10.

Step 3: Isolate the Variable

Now, to finish isolating 'x', divide both sides of the inequality by the coefficient of 'x' which is 2. Therefore, x < -10 / 2 and solving that gives the result: x < -5.

Key Concepts

Isolating Variables in Inequalities

In solving inequalities, one of the key steps is to isolate the variable. This means rearranging the inequality so that the variable you are solving for is on one side, preferably by itself.

• First, identify the term on the same side of the inequality symbol as the variable.
• Perform arithmetic operations such as addition, subtraction, multiplication, or division to move all other terms to the opposite side of the inequality.
• In our example, after removing the negative sign, the inequality becomes \(2x + 4 < -6\).

Next, we subtract 4 from both sides. This leaves us with \(2x < -10\), effectively reducing the inequality to a simpler form. Remember to perform the same operation on both sides to keep the inequality balanced.
When you isolate the variable, you make it easier to see what values satisfy the inequality.

Multiplying or Dividing by a Negative Number

When you multiply or divide both sides of an inequality by a negative number, it's crucial to reverse the direction of the inequality symbol. This is a fundamental rule because multiplying or dividing by a negative flips the number line.
In our exercise, the inequality starts with a negative sign in front of the parentheses, \(-(2x + 4) > 6\).

• To remove the negative sign, we multiply both sides by -1.
• This changes the direction of the greater than symbol to a less than symbol.
• The inequality becomes \(2x + 4 < -6\).

Always remember this rule:

• Multiply by a negative, flip the inequality.
• Divide by a negative, flip the inequality.

This keeps your solution accurate and reflects the true sense of the inequality.

Reversing the Inequality Sign

Reversing the inequality sign is an integral part of solving inequalities, especially when dealing with multiplication or division by negative numbers.

• Think of the inequality like a balance scale. Flipping the direction happens when you change the sign of what is weighted on one side.
• As demonstrated in the exercise, \(-1\) was multiplied to both sides, which reversed \(>\) to \(<\).
• This ensures that the inequality still maintains its truth.

Reversing the sign can seem tricky at first, but with practice, it becomes second nature. Keeping track of these changes is essential to reaching the correct solution. Always ask yourself: was a negative factor involved in my step? If so, remember to flip the inequality.