Just as equations are mathematical sentences which state that two quantities are equal, inequalities are sentences which state that two quantites are not equal (or possibly not equal). There are five inequality symbols:
Often, for the last four types of inequalities, we need to solve the inequality so that the variable is alone on one side. This is done using analogues of the properties of equality: adding or subtracting the same quantity to both sides, or multiplying or dividing both sides by the same quantity. The only important difference is that:
Solve for x.
–3x + 2 14
First, subtract 2 from both sides.
Now divide both sides by –3. Remember to reverse the inequality.
The solutions of inequalities can be graphed on the number line as rays. If the inequality is "strict" (< or >), we use an open dot to indicate that the endpoint of the ray is not part of the solution. For the other types of inequalities ( and ), we use a closed dot.
Graph the inequality.
Solve the inequality and graph its solution.
x + 6 8
–4 + f 20
p – 12 –1
x + 7 –13
7m – 6m 3
Find the approximate solution and graph it.
Solve the inequality. Then match its solution with one of the graphs shown.
Arfana had $100 to buy gifts for her family. She bought a book for $30.75 for her father, a $20.50 necklace for her mother, and a CD for $18.42 for her brother. She still has to buy a present for her sister. Write and solve an inequality to express how much she can spend on her sister's gift.
Three consecutive integers can be expressed as m,m + 1, and m + 2. Write an inequality, and use it to find the smallest consecutive integers satisfying the condition that the sum is greater than 61.
A couple goes out to dinner with $50 to spend. A sales tax of 5% is added to the bill, and they plan to tip 20% after tax. Write and solve an inequality to show how much they can spend on the meal.