\( 4x + 5\left( x + 6 \right) - 10 \lt 9 \)
\( 9x + 20 \lt 9 \)
Distribute and then collect the like terms on the left-hand side of the inequality.
\( 4x + 5\left( x + 6 \right) - 10 \lt 9 \)
\( 4x + 5x + 30 - 10 \lt 9 \)
\( 9x + 20 \lt 9 \)
Let's have a look...
You encountered the concept of like terms earlier in this course. Remember that terms are single numbers, variables, or a number that is paired with a variable. Like terms are those that have the same variable raised to the same exponent.
Examples of Like Terms
Examples of like terms include:
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\( \frac{3}{4} \) and \( 7 \)
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\( 2x \) and \( - 5x \)
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\( 9.2g^{2} \) and \( - 4.7g^{2} \)
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\( -qr \) and \( 8qr \)
You have learned that when terms have the same variable raised to the same exponent, you can simplify your expression or equation by combining the coefficients. The terms that have the same variable(s) are referred to as like terms, and the process of simplifying is referred to as combining like terms.
While you have already worked with like terms in expressions and equations, inequalities can also contain like terms.
For Example
Collect the like terms, if any, in the inequality \( 3x + 4\left( x - 2 \right) + 1 \gt 0 \).
There are \( x \)-terms on the left-hand side of the inequality. To collect them, you will need to apply the distributive property first. |
\( 3x + 4\left( x - 2 \right) + 1 \gt 0 \) \( 3x + 4x - 8 + 1 \gt 0 \) |
Collect the like terms on the left-hand side of the inequality. |
The terms \( 3x \) and \( 4x \) are like. \( \color{#A80000}{3x + 4x} - 8 + 1 \gt 0 \) \( \color{#A80000}{7x} - 8 + 1 \gt 0 \) The terms \( - 8 \) and \( 1 \) are like. \( 7x \color{#A80000}{- 8 + 1} \gt 0 \) \( 7x \color{#A80000}{- 7} \gt 0 \) |
Let's Practice
Complete the activity below to practice identifying and collecting like terms in inequalities. Determine whether the equality on each slide has like terms. If there are no like terms, then say so. If there are like terms, simplify the inequality by collecting the like terms. Be sure to check your answer.
