Question

Solve the inequality $10 - x \geqslant 6$?

Answer 1

Inequalities are used to compare the two values, showing if one is greater than, less than or simply not equal to another. These values could be numerical or algebraic or a combination of both. The algebraic inequalities are also called literal inequalities. We use different types of symbols to compare the values. The symbol $\leqslant$ represent less than or equal to, $\geqslant$ represent greater than or equal to, $\ne$ represent not equal to, > represent greater than, < represent less than. A linear inequality is like a linear equation. Inequalities that have the same solutions are called equivalent.

Complete step by step solution:
The given inequality is,
$10 - x \geqslant 6$
Subtract 10 from both sides of the inequality and solve.
$\Rightarrow 10 - x - 10 \geqslant 6 - 10 \\\ \Rightarrow - x \geqslant - 4 \\\$
Multiply with $\left( { - 1} \right)$to both sides of the inequality and reverse the sign of inequality.
$\Rightarrow \left( { - x} \right) \times \left( { - 1} \right) \geqslant \left( { - 4} \right) \times \left( { - 1} \right) \\\ \Rightarrow x \geqslant 4 \\\ \Rightarrow x \leqslant 4 \\\$

Therefore the solution of the inequality is $x \leqslant 4$.

Additional information:
Properties of inequality:
• The additional property of inequality is defined as; when we add the same number to both sides of the inequality it produces an equivalent inequality.
• The subtraction property of the inequality is defined as; when we subtract the same number from both sides of the inequality it gives an equivalent inequality.
• The multiplication property of the inequality says that multiplying with a positive number to the both sides of the inequality produces an equivalent inequality.
• Multiplication with a negative number to the both sides of the inequality will not produce an equivalent inequality. When we multiply an inequality with a negative number to both sides of the inequality then it reverses its direction of the inequality.

Note: Reverse the direction of the sign of inequality when multiplying with a negative number to both sides of the inequality. Addition of a positive number does not change the direction of the inequality. When solving a multi step inequality always remember to change the direction of inequality sign while multiplying or dividing with a negative number.