Secondary II • 28d

how can I solve for x in inequalities

How can I solve for x in inequalities

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Secondary II • 28d

how can I solve for x in inequalities

How can I solve for x in inequalities

Explanation from Alloprof

This Explanation was submitted by a member of the Alloprof team.

Hi!

To solve an inequality, you must always place like terms on one side of the inequality, and constants on the other side. Let's take an example to better understand.

We have the inequality:

$$ 4x - 6 < 2x + 10$$

Like terms are terms that have the same variables (the same unknowns), and these variables are assigned the same exponents. Our like terms here are \(4x \) and \( 2x\), since they both contain the variable x which has an exponent 1.

Constants are terms that do not contain variables, here \(-6\) and \(10\).

Our goal will first be to place the two like terms on one side of the inequality, and the constants on the other side. To do this, we will start by moving one of the two like terms to the other side (it does not matter which one).

Let's move \(2x\) to the left side of the inequality. Since the inverse of addition is subtraction, we'll need to subtract \(2x\) from each side of the inequality, like this:

$$ 4x - 6 -2x< 2x + 10-2x$$

By subtracting it from each side, this allows us to eliminate it from the right side of the inequality:

$$ 4x - 6 -2x< 10$$

We thus moved the term \(2x\) so that it is on the same side as \(4x\).

Now, let's move on to constants. We will move the constant \(6\) to the other side. Since the inverse operation of a subtraction is an addition, we will therefore add \(6\) on each side:

$$ 4x - 6 -2x+6< 10+6$$

$$ 4x -2x< 10+6$$

We succeeded in placing our like terms on one side and our constants on the other! The next step will be to add the constants, and add the coefficients of like terms. Let's start with the constants. Since 10+6 gives 16, we have:

$$ 4x -2x< 16$$

To subtract like terms, you must subtract their coefficient, which is the number in front of the variable x.

$$ (4-2)x< 16$$

$$ 2x< 16$$

Finally, the last step will be to eliminate the coefficient of the variable x, i.e. \(2\), by performing the opposite operation of multiplication, i.e. division:

$$ \frac{2x}{2} < \frac{16}{2} $$

$$x< 8$$

Here are some sheets on these concepts that might be useful to you:

I hope I was able to help you! Feel free to ask us more questions if you have any! :)