INEQUALITIES Math 7

Symbols < less than > greater than ≤ less than or equal to ≥ greater than or equal to

Meaning of Symbols x > 2 means that the value of x is greater than 2 (but does not include 2) x ≥ 2 means that the value of x is greater than or equal to 2 (includes 2 and anything larger) x < 2 means that the value of x is less than 2 (but does not include 2) x ≤ 2 means that the value of x is less than or equal to 2 (includes 2 and anything smaller)

Examples x > 4 What could x be? x < 9 What could x be?

INEQUALITIES IN WORDS Sometimes, you will have to read a statement to determine which inequality to use. Read the statement carefully because these can get tricky! When you see words like “at least” or “at most”, how do you handle them? For example: At least 5 people were late today The stadium holds at most 48,000 people

BE CAREFUL! No more than 50 students Fewer than 12 items Up to and including 12 *The speed limit on a road is 55 mph*

BACKWARDS? x > 9 is read “x is greater than 9” What does this say about 9? So, if you had 12 > x, what does that say about x? If they are backwards, you don’t have to panic! 13 < x would mean what?

TRANSLATE SOME the number of students is greater than 13 the amount of money is less than or equal to $50 60 is greater than or equal to the number the perimeter is no less than 85 inches the area is no greater than 100 square cm fewer than 12 items at least 275 people attended the play at most $50 for admission to the game

Graphing Inequalities These are graphed on number lines. Be sure the problem is solved for x. Draw a number line with x, some values smaller than x, and some larger. Go to x on the number line and use: ○ if the number “is not included in the solution” (symbol is < or >) ● if the number “is included in the solution” (symbol is ≤ or ≥)

Graphing Inequalities The last step is to draw an arrow pointing in the proper direction. “less than” should be pointed in direction that has numbers less than x “greater than” should be pointed in the direction that has numbers greater than x

Examples x > 5 x < 9 x ≥ 6 x ≤ 3

Examples x < -7 x > -3 x ≤ -5 x ≥ -8

CAREFUL WITH THE BACKWARDS EXAMPLES

Solving One-Step Inequalities Sometimes these will not be in the form “x > ..” In this case, we will have to solve for x. Example: x – 9 > 5 Solve for x as if the > were an = (inverse operations). Check to make sure it creates a true statement When you check, make sure to choose a number that is described by the inequality

Examples CHECKS x – 9 < 11 x + 12 > 9 12 ≥ -18 + x -11 ≥ x – 18

TRANSLATE, SOLVE , GRAPH The sum of t and 9 is greater than or equal to 36 7 more than w is less than or equal to 10 19 increased by p is greater than or equal to 2 Three plus a number is at least six

Examples 5x > -40 3x < 12 5x ≤ -30 7x ≤ -56

DIVIDING BY A NEGATIVE When solving, dividing by a negative changes the game of the inequality. Let’s see what happens here: -7x > 63 To solve, you must divide by -7 on each side x > -9 Choose a number to plug this back into the original to see if it “checks” -7(-8) > 63? Is this true? Ok, try -7 (since -7 > -9) -7(-7) > 63? Is this true? So try something smaller than -9…like -10.

DIVIDING BY A NEGATIVE -10 did work…weird right? Basically, whenever you divide by a negative, the inequality needs to be flipped around to the opposite (less than goes to greater than and vice versa; less than or equal to goes to greater than or equal to and vice versa) This only happens when the number you are DIVIDING BY is negative This will also happen if you multiply both sides by a negative (like when we use the multiplicative inverse to clear a fraction)

MORE EXAMPLES Solve and Graph: -9x > 90 -11x < -99 12x ≤ -36

TRANSLATE, SOLVE , GRAPH Double a number is at most four 6 is less than the product of f and 20 A number divided by -3 is at least seven

TWO-STEP INEQUALITIES Treat these the same way you would treat a two-step equation Take care of the addition/subtraction part first Then move on to the multiplication/division portion Remember, if you multiply or divide BY a negative, you must flip the symbol

SOLVE THE INEQUALITIES 5x – 4 > -24 6 + 3x < -27 -9y + 33 ≥ 51