Mathematics 4 Life and Beyond: Solving Multi-Step Inequalities (12-7)

Solving Multi-Step Inequalities

Examples

Solving multi-step inequalities is very similar to solving equations—what you do to one side you need to do to the other side in order to maintain the balance” of the inequality. The Properties of Inequality can help you understand how to add, subtract, multiply, or divide within an inequality.

Using Properties Together to Solve Inequalities

A popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other. As with one step inequalities, the solutions to multi-step inequalities can be graphed on a number line.

Example
Problem	Solve for p. 4p + 5 < 29
		Begin to isolate the variable by subtracting 5 from both sides of the inequality. Divide both sides of the inequality by 4 to express the variable with a coefficient of 1.
Answer

To graph this inequality, you draw an open circle at the end point 6 on the number line. The circle is open because the inequality is less than 6 and not equal to 6. The values where p is less than 6 are found all along the number line to the left of 6. Draw a blue line with an arrow on the number line pointing in that direction.

To check the solution, substitute the end point 6 into the original inequality written as an equation, which is called the related equation, to see if you get a true statement. Then check another solution, such as 0, to see if the inequality is correct.

Example
Problem	Check that p < 6 is the solution to the inequality 4p + 5 < 29.
		Check the end point 6 in the related equation.
		Try another value to check the inequality. Let’s use p = 0.
Answer p < 6 is the solution to the inequality 4p + 5 < 29.

Example
Problem	Solve for x. 3x – 7≥ 41
		Begin to isolate the variable by adding 7 to both sides of the inequality. Divide both sides of the inequality by 3 to express the variable with a coefficient of 1.
Check		First, check the end point 16 in the related equation. Then, try another value to check the inequality. Let’s use x = 20.
Answer

When solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.

Example
Problem	Solve for p. −6p + 14 < −58
		Begin to isolate the variable by subtracting 14 from both sides of the inequality. Divide both sides of the inequality by −6 to express the variable with a coefficient of 1. Dividing by a negative number results in reversing the inequality sign.
Check		Check the solution. First, check the end point 12 in the related equation. Then, try another value to check the inequality. Try 100.
Answer

The graph of the inequality p > 12 has an open circle at 12 with an arrow stretching to the right.

Advanced Example
Problem	Solve for x.
		To isolate the variable, subtract from both sides of the inequality. Then multiply by 3 so that the coefficient in front of the parentheses is 1. Then subtract 3 from both sides.
Check		Check the solution. First, check the end point -18 in the related equation.
		Now check any value for x that is within the region . We will use . The statement is true.
Answer

Advanced Question

A student is solving the inequality . If she combines like terms, which of the following inequalities could she see?

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Using the Distributive Property to Clear Parentheses and Fractions

As with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.

Example
Problem	Solve for x. 2(3x – 5) ≤ 4x + 6
		Distribute to clear the parentheses. Subtract 4x from both sides to get the variable term on one side only. Add 10 to both sides to isolate the variable. Divide both sides by 2 to express the variable with a coefficient of 1.
Check		Check the solution. First, check the end point 8 in the related equation. Then, choose another solution and evaluate the inequality for that value to make sure it is a true statement. Try 0.
Answer

Example
Problem	Solve for a.
		Clear the fraction by multiplying both sides of the equation by 6. Add 4 to both sides to isolate the variable. Divide both sides by 2 to express the variable with a coefficient of 1.
Check		Check the solution. First, check the end point 8 in the related equation. Then, choose another solution and evaluate the inequality for that value to make sure it is a true statement. Try 5.
Answer

Advanced Example
Problem	Solve for d.
		This inequality contains two parentheses. Use the Distributive Property to expand both sides of the inequality.
		Now that both sides have been expanded, combine like terms and find the range of values for d.
Check		Check the solution. First, check the end point in the related equation. It results in a true statement.
		Now try any value for d that is within the region . We will try This is also a true statement.
Answer