Inequalities on the ACT and SAT
We can break SAT and ACT inequalities questions down into three types: those involving a word problem, those involving algebra and those involving absolute values. Let’s tackle the word problems first.
A pulley can handle no more than 800 lbs of weight. It is currently holding 4 steel frames that weigh 112 lbs each. Amanda wants to load as many bricks onto it as she can without it breaking. If x represents the total weight of bricks, in lbs, that she can add, which of the following inequalities could be used to determine possible values of x?
From the question, you can deduce the following information:
• You can have a MAXIMUM of 800 lbs
• You have 4 frames, each weighing 112 lbs. That means, all 4 frames weight 4*112 = 448 lbs
• In addition to the 4 frames, you want to load x lbs in bricks.
You can thus conclude that x + 448 lbs must not exceed 800 lbs or my pulley will break.
In mathematical notation, x + 4(112) < 800 Don’t get tricked if the answer is written as 800 > x + 4(112). This statement is exactly the same as the statement above. 800 is greater than x + 4(112) is the same as x + 4(112) is less than 800.
Another type of inequality involves algebra. Suppose y = 2x + 4 and x < 3 and you need to find an inequality involving y.
- Start with the inequality you have x < 3
- Look at the equation involving y. There is a 2x in it. Since x < 3, that means 2x < 6
- If 2x < 6, that means 2x + 4 < 6 + 4 = 10. Thus 2x + 4 < 10
- But 2x + 4 is simply y, so we can conclude that y < 10
The trick is to make your inequality look like the equation. Can you work the next two examples out on your own? Don’t forget, that when multiplying an inequality by a negative number, you have to switch the signs.
1. If 2y = 2x + 4 and x < 3, find an inequality involving y
2. If y = -2x + 4 and x < 3, find an inequality involving y
Don’t forget, that when multiplying an inequality by a negative number, you have to switch the signs.
Like before, we have 2x < 6 and 2x + 4 < 10. But now we have 2y < 10, meaning y < 5
If x < 3 , then -2x > -6. That means that -2x + 4 > -2 so we get y > -2
The last type of question tests your knowledge of absolute values. Absolute values are denoted by two straight lines | |. Absolute values make negative numbers positive.
So |10| = 10 and |-10| = 10 too.
Now that we’ve established what absolute values do, we can solve absolute value inequalities.
Practice Absolute Value Inequality Question
A theater company is auditioning for actors to portray the leading character in a new play. The company is looking for actors between the ages of 20 and 40 (inclusive). Which of the following inequalities can be used to determine whether an actor of age a is eligible to audition for the part?
- | a-10 | ≤ 40
- | a-20 | ≤ 40
- | a-30 | ≤ 20
- | a-30 | ≤ 10
- | a-35 | ≤ 5
The answer is 20 ≤ a ≤ 40. But how do we make that look like one of the inequalities above?
- Take the average of 20 and 40. That’s 30.
- Subtract 30 from everything to get 20 – 10 ≤ a – 30 ≤ 40 – 30
- You get -10 ≤ a – 30 ≤ 10 Note that the left and right side of the inequality is the same number, except that the one of the left is negative
- Now we can say, | a – 30 | ≤ 10
So the answer is clearly choice D.
Can you work out the lower and upper bounds of the other choices? Choice A is solved below for you.
Choice A says | a – 10 | ≤ 40.
=> -40 ≤ a – 10 ≤ 40
=> -40 + 10 £ a – 10 + 10 £ 40 + 10
Thus Choice A is effectively saying -30 ≤ a ≤ 50, which is not the answer at all.