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# How do you solve the inequality $4x - 2 < 6$ or $3x + 1 > 22$?

Last updated date: 19th Sep 2024
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Hint: An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. Here we need to solve for ‘x’ which is a variable. Solving the given inequality is very like solving equations and we do most of the same thing but we must pay attention to the direction of inequality$( \leqslant , > )$. We have two linear inequalities. We can solve this.

Complete step-by-step solution:
Given, $4x - 2 < 6$
We add 2 on both sides and we know that the inequality direction doesn’t change,
$4x - 2 + 2 < 6 + 2 \\ 4x < 8 \\$
We divide 4 on both sides we have,
$x < \dfrac{8}{4} \\ x < 2 \\$
Thus the solution of $4x - 2 < 6$ is $x < 2$. The interval form is $( - \infty ,2)$.
Now take $3x + 1 > 22$
Subtract 1 on both sides we have,
$3x + 1 - 1 > 22 - 1 \\ 3x > 21 \\$
Divide by 3 on both sides we have,
$x > \dfrac{{21}}{3} \\ x > 7 \\$
Thus the solution of $3x + 1 > 22$ is $x > 7$. The interval form is $(7,\infty )$.

Note: We take value of ‘x’ in $( - \infty ,2)$ and put it in $4x - 2 < 6$
Let’s put $x = 0$ in $4x - 2 < 6$
$4(0) - 2 < 6 \\ - 2 < 6 \\$
Which is correct. We check for the second inequality in the same way.
We know that $a \ne b$ says that ‘a’ is not equal to ‘b’. $a > b$ means that ‘a’ is less than ‘b’. $a < b$ means that ‘a’ is greater than ‘b’. These two are known as strict inequality. $a \geqslant b$ means that ‘a’ is less than or equal to ‘b’. $a \leqslant b$ means that ‘a’ is greater than or equal to ‘b’.

The direction of inequality do not change in these cases:
-Add or subtract a number from both sides.
-Multiply or divide both sides by a positive number.
-Simplify a side.

The direction of the inequality change in these cases:
-Multiply or divide both sides by a negative number.
-Swapping left and right hand sides.